UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549


                                  SCHEDULE 13G

                    Under the Securities Exchange Act of 1934
                                (Amendment No. )*

                          Alexion Pharmaceuticals, Inc.
--------------------------------------------------------------------------------
                                (Name of Issuer)

                    Common Stock, $.0001 Par Value Per Share
--------------------------------------------------------------------------------
                         (Title of Class of Securities)

                                    01531109
                                    --------
                                 (CUSIP Number)

                                December 29, 2004
                                -----------------
                      (Date of Event which Requires Filing
                               of this Statement)



Check the appropriate box to designate the rule pursuant to which this Schedule
                                    is filed:

                                [ ] Rule 13d-1(b)
                                [X] Rule 13d-1(c)
                                [ ] Rule 13d-1(d)



*The remainder of this cover page shall be filled out for a reporting person's
initial filing on this form with respect to the subject class of securities, and
for any subsequent amendment containing information which would alter the
disclosures provided in a prior cover page.

The information required on the remainder of this cover page shall not be deemed
to be "filed" for the purpose of Section 18 of the Securities Exchange Act of
1934 ("Act") or otherwise subject to the liabilities of that section of the Act
but shall be subject to all other provisions of the Act (however, see the
Notes).


                                  Page 1 of 13




------------------                                            ------------------
CUSIP No. 01531109                    13G                     Page 2 of 13 Pages
------------------                                            ------------------

---------- ---------------------------------------------------------------------
        1  NAME OF REPORTING PERSON
           I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

           S.A.C. Capital Advisors, LLC
---------- ---------------------------------------------------------------------
        2  CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*
                                                                         (a) [ ]

                                                                         (b) [X]

---------- ---------------------------------------------------------------------
        3  SEC USE ONLY
---------- ---------------------------------------------------------------------
        4  CITIZENSHIP OR PLACE OF ORGANIZATION

           Delaware
--------------------- ----- ----------------------------------------------------
                         5  SOLE VOTING POWER

                            0
NUMBER OF             ----- ----------------------------------------------------
SHARES                   6  SHARED VOTING POWER
BENEFICIALLY
OWNED                       38,150 (see Item 4)
BY                    ----- ----------------------------------------------------
EACH                     7  SOLE DISPOSITIVE POWER
REPORTING
PERSON                      0
WITH                  ----- ----------------------------------------------------
                         8  SHARED DISPOSITIVE POWER

                            38,150 (see Item 4)
---------- ---------------------------------------------------------------------
        9  AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

           38,150 (see Item 4)
---------- ---------------------------------------------------------------------
       10  CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES

           [ ]

---------- ---------------------------------------------------------------------
       11  PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)

           Less than 0.1% (see Item 4)
---------- ---------------------------------------------------------------------
       12  TYPE OF REPORTING PERSON*

           00
---------- ---------------------------------------------------------------------
                       *SEE INSTRUCTION BEFORE FILLING OUT


                                  Page 2 of 13




------------------                                            ------------------
CUSIP No. 01531109                    13G                     Page 3 of 13 Pages
------------------                                            ------------------

---------- ---------------------------------------------------------------------
        1  NAME OF REPORTING PERSON
           I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

           S.A.C. Capital Management, LLC
---------- ---------------------------------------------------------------------
        2  CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*
                                                                         (a) [ ]

                                                                         (b) [X]

---------- ---------------------------------------------------------------------
        3  SEC USE ONLY
---------- ---------------------------------------------------------------------
        4  CITIZENSHIP OR PLACE OF ORGANIZATION

           Delaware
--------------------- ----- ----------------------------------------------------
                         5  SOLE VOTING POWER

                            0
NUMBER OF             ----- ----------------------------------------------------
SHARES                   6  SHARED VOTING POWER
BENEFICIALLY
OWNED                       38,150 (see Item 4)
BY                    ----- ----------------------------------------------------
EACH                     7  SOLE DISPOSITIVE POWER
REPORTING
PERSON                      0
WITH                  ----- ----------------------------------------------------
                         8  SHARED DISPOSITIVE POWER

                            38,150 (see Item 4)
---------- ---------------------------------------------------------------------
        9  AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

           38,150 (see Item 4)
---------- ---------------------------------------------------------------------
       10  CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES

           [ ]

---------- ---------------------------------------------------------------------
       11  PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)

           Less than 0.1% (see Item 4)
---------- ---------------------------------------------------------------------
       12  TYPE OF REPORTING PERSON*

           00
---------- ---------------------------------------------------------------------
                       *SEE INSTRUCTION BEFORE FILLING OUT


                                  Page 3 of 13




------------------                                            ------------------
CUSIP No. 01531109                    13G                     Page 4 of 13 Pages
------------------                                            ------------------

---------- ---------------------------------------------------------------------
        1  NAME OF REPORTING PERSON
           I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

           S.A.C. MultiQuant Fund, LLC
---------- ---------------------------------------------------------------------
        2  CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*
                                                                         (a) [ ]

                                                                         (b) [X]

---------- ---------------------------------------------------------------------
        3  SEC USE ONLY
---------- ---------------------------------------------------------------------
        4  CITIZENSHIP OR PLACE OF ORGANIZATION

           Anguilla, British West Indies
--------------------- ----- ----------------------------------------------------
                         5  SOLE VOTING POWER

                            0
NUMBER OF             ----- ----------------------------------------------------
SHARES                   6  SHARED VOTING POWER
BENEFICIALLY
OWNED                       38,150 (see Item 4)
BY                    ----- ----------------------------------------------------
EACH                     7  SOLE DISPOSITIVE POWER
REPORTING
PERSON                      0
WITH                  ----- ----------------------------------------------------
                         8  SHARED DISPOSITIVE POWER

                            38,150 (see Item 4)
---------- ---------------------------------------------------------------------
        9  AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

           38,150 (see Item 4)
---------- ---------------------------------------------------------------------
       10  CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES

           [ ]

---------- ---------------------------------------------------------------------
       11  PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)

           Less than 0.1% (see Item 4)
---------- ---------------------------------------------------------------------
       12  TYPE OF REPORTING PERSON*

           00
---------- ---------------------------------------------------------------------
                       *SEE INSTRUCTION BEFORE FILLING OUT


                                  Page 4 of 13




------------------                                            ------------------
CUSIP No. 01531109                    13G                     Page 5 of 13 Pages
------------------                                            ------------------

---------- ---------------------------------------------------------------------
        1  NAME OF REPORTING PERSON
           I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

           Sigma Capital Management, LLC
---------- ---------------------------------------------------------------------
        2  CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*
                                                                         (a) [ ]

                                                                         (b) [X]

---------- ---------------------------------------------------------------------
        3  SEC USE ONLY
---------- ---------------------------------------------------------------------
        4  CITIZENSHIP OR PLACE OF ORGANIZATION

           Delaware
-------------------- ------ ----------------------------------------------------
                         5  SOLE VOTING POWER

                            0
NUMBER OF            ------ ----------------------------------------------------
SHARES                   6  SHARED VOTING POWER
BENEFICIALLY
OWNED                       1,471,520 (see Item 4)
BY                   ------ ----------------------------------------------------
EACH                     7  SOLE DISPOSITIVE POWER
REPORTING
PERSON                      0
WITH                 ------ ----------------------------------------------------
                         8  SHARED DISPOSITIVE POWER

                            1,471,520 (see Item 4)
---------- ---------------------------------------------------------------------
        9  AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

           1,471,520 (see Item 4)
---------- ---------------------------------------------------------------------
       10  CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES

           [ ]

---------- ---------------------------------------------------------------------
       11  PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)

           5.3% (see Item 4)
---------- ---------------------------------------------------------------------
       12  TYPE OF REPORTING PERSON*

           00
---------- ---------------------------------------------------------------------
                       *SEE INSTRUCTION BEFORE FILLING OUT


                                  Page 5 of 13




------------------                                            ------------------
CUSIP No. 01531109                    13G                     Page 6 of 13 Pages
------------------                                            ------------------

---------- ---------------------------------------------------------------------
        1  NAME OF REPORTING PERSON
           I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

           Sigma Capital Associates, LLC
---------- ---------------------------------------------------------------------
        2  CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*
                                                                         (a) [ ]

                                                                         (b) [X]

---------- ---------------------------------------------------------------------
        3  SEC USE ONLY
---------- ---------------------------------------------------------------------
        4  CITIZENSHIP OR PLACE OF ORGANIZATION

           Anguilla, British West Indies
-------------------- ------ ----------------------------------------------------
                         5  SOLE VOTING POWER

                            0
NUMBER OF            ------ ----------------------------------------------------
SHARES                   6  SHARED VOTING POWER
BENEFICIALLY
OWNED                       1,471,520 (see Item 4)
BY                   ------ ----------------------------------------------------
EACH                     7  SOLE DISPOSITIVE POWER
REPORTING
PERSON                      0
WITH                 ------ ----------------------------------------------------
                         8  SHARED DISPOSITIVE POWER

                            1,471,520 (see Item 4)
---------- ---------------------------------------------------------------------
        9  AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

           1,471,520 (see Item 4)
---------- ---------------------------------------------------------------------
       10  CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES

           [ ]

---------- ---------------------------------------------------------------------
       11  PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)

           5.3% (see Item 4)
---------- ---------------------------------------------------------------------
       12  TYPE OF REPORTING PERSON*

           00
---------- ---------------------------------------------------------------------
                       *SEE INSTRUCTION BEFORE FILLING OUT


                                  Page 6 of 13




------------------                                            ------------------
CUSIP No. 01531109                    13G                     Page 7 of 13 Pages
------------------                                            ------------------

---------- ---------------------------------------------------------------------
        1  NAME OF REPORTING PERSON
           I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

           Steven A. Cohen
---------- ---------------------------------------------------------------------
        2  CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*
                                                                         (a) [ ]

                                                                         (b) [X]

---------- ---------------------------------------------------------------------
        3  SEC USE ONLY
---------- ---------------------------------------------------------------------
        4  CITIZENSHIP OR PLACE OF ORGANIZATION

           United States
-------------------- ------ ----------------------------------------------------
                         5  SOLE VOTING POWER

                            0
NUMBER OF            ------ ----------------------------------------------------
SHARES                   6  SHARED VOTING POWER
BENEFICIALLY
OWNED                       1,509,670 (see Item 4)
BY                   ------ ----------------------------------------------------
EACH                     7  SOLE DISPOSITIVE POWER
REPORTING
PERSON                                  0
WITH                 ------ ----------------------------------------------------
                         8  SHARED DISPOSITIVE POWER

                            1,509,670 (see Item 4)
---------- ---------------------------------------------------------------------
        9  AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

           1,509,670 (see Item 4)
---------- ---------------------------------------------------------------------
       10  CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES



---------- ---------------------------------------------------------------------
       11  PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)

           5.4% (see Item 4)
---------- ---------------------------------------------------------------------
       12  TYPE OF REPORTING PERSON*

           IN
---------- ---------------------------------------------------------------------
                       *SEE INSTRUCTION BEFORE FILLING OUT


                                  Page 7 of 13




Item 1(a)           Name of Issuer:
                    ---------------

                    Alexion Pharmaceuticals, Inc.

Item 1(b)           Address of Issuer's Principal Executive Offices:
                    ------------------------------------------------

                    352 Knotter Drive
                    Cheshire, Connecticut 06410

Items 2(a)          Name of Person Filing:

                    This statement is filed by: (i) S.A.C. Capital Advisors,
                    LLC, ("SAC Capital Advisors") with respect to shares of
                    common stock, $.0001 par value per share ("Shares") of the
                    Issuer beneficially owned by S.A.C. MultiQuant Fund, LLC
                    ("SAC MultiQuant"); (ii) S.A.C. Capital Management, LLC,
                    ("SAC Capital Management") with respect to Shares
                    beneficially owned by SAC MultiQuant; (iii) SAC MultiQuant
                    with respect to Shares beneficially owned by it; (iv) Sigma
                    Capital Management, LLC ("Sigma Capital Management") with
                    respect to Shares beneficially owned by Sigma Capital
                    Associates, LLC ("Sigma Capital Associates"); (v) Sigma
                    Capital Associates with respect to Shares beneficially owned
                    by it; and (vi) Steven A. Cohen with respect to Shares
                    beneficially owned by SAC Capital Advisors, SAC Capital
                    Management, SAC MultiQuant, Sigma Capital Management and
                    Sigma Capital Associates.

Item 2(b)           Address of Principal Business Office:
                    -------------------------------------

                    The address of the principal business office of (i) SAC
                    Capital Advisors and Mr. Cohen is 72 Cummings Point Road,
                    Stamford, Connecticut 06902, (ii) SAC Capital Management and
                    Sigma Capital Management is 540 Madison Avenue, New York,
                    New York 10022, and (iii) SAC MultiQuant and Sigma Capital
                    Associates is P.O. Box 58, Victoria House, The Valley,
                    Anguilla, British West Indies.

Item 2(c)           Citizenship:
                    ------------

                    SAC Capital Advisors, SAC Capital Management and Sigma
                    Capital Management are Delaware limited liability companies.
                    SAC MultiQuant and Sigma Capital Associates are Anguillan
                    limited liability companies. Mr. Cohen is a United States
                    citizen.


                                  Page 8 of 13




Item 2(d)           Title of Class of Securities:
                    -----------------------------

                    Common Stock, par value $.0001 per share

Item 2(e)           CUSIP Number:
                    -------------

                    01531109

Item 3              Not Applicable

Item 4              Ownership:
                    ----------

                    The percentages used herein are calculated based upon the
                    Shares issued and outstanding as of November 30, 2004 as
                    reported on the Issuers's quarterly report on Form 10-Q
                    filed with the Securities and Exchange Commission by the
                    Company for the quarterly period ended October 31, 2004.

                    As of the close of business on January 6, 2005:

                    1. S.A.C. Capital Advisors, LLC
                    (a) Amount beneficially owned: 38,150
                    (b) Percent of class: Less than 0.1%
                    (c)(i) Sole power to vote or direct the vote: -0-
                    (ii) Shared power to vote or direct the vote: 38,150
                    (iii) Sole power to dispose or direct the disposition: -0-
                    (iv) Shared power to dispose or direct the disposition:
                    38,150

                    2. S.A.C. Capital Management, LLC
                    (a) Amount beneficially owned: 38,150
                    (b) Percent of class: Less than 0.1%
                    (c)(i) Sole power to vote or direct the vote: -0-
                    (ii) Shared power to vote or direct the vote: 38,150
                    (iii) Sole power to dispose or direct the disposition: -0-
                    (iv) Shared power to dispose or direct the disposition:
                    38,150

                    3. S.A.C. MultiQuant Fund, LLC
                    (a) Amount beneficially owned: 38,150
                    (b) Percent of class: Less than 0.1%
                    (c)(i) Sole power to vote or direct the vote: -0-
                    (ii) Shared power to vote or direct the vote: 38,150
                    (iii) Sole power to dispose or direct the disposition: -0-
                    (iv) Shared power to dispose or direct the disposition:
                    38,150


                                  Page 9 of 13




                    4. Sigma Capital Management, LLC
                    (a) Amount beneficially owned: 1,471,520
                    (b) Percent of class: 5.3%
                    (c)(i) Sole power to vote or direct the vote: -0-
                    (ii) Shared power to vote or direct the vote: 1,471,520
                    (iii) Sole power to dispose or direct the disposition: -0-
                    (iv) Shared power to dispose or direct the disposition:
                    1,471,520

                    5. Sigma Capital Associates, LLC
                    (a) Amount beneficially owned: 1,471,520
                    (b) Percent of class: 5.3%
                    (c)(i) Sole power to vote or direct the vote: -0-
                    (ii) Shared power to vote or direct the vote: 1,471,520
                    (iii) Sole power to dispose or direct the disposition: -0-
                    (iv) Shared power to dispose or direct the disposition:
                    1,471,520

                    6. Steven A. Cohen
                    (a) Amount beneficially owned: 1,509,670
                    (b) Percent of class: 5.4%
                    (c)(i) Sole power to vote or direct the vote: -0-
                    (ii) Shared power to vote or direct the vote: 1,509,670
                    (iii) Sole power to dispose or direct the disposition: -0-
                    (iv) Shared power to dispose or direct the disposition:
                    1,509,670

                    SAC Capital Advisors, SAC Capital Management, Sigma Capital
                    Management, and Mr. Cohen own directly no Shares. Pursuant
                    to investment agreements, each of SAC Capital Advisors and
                    SAC Capital Management share all investment and voting power
                    with respect to the securities held by SAC MultiQuant.
                    Pursuant to an investment management agreement, Sigma
                    Capital Management maintains investment and voting power
                    with respect to the securities held by Sigma Capital
                    Associates. Mr. Cohen controls each of SAC Capital Advisors,
                    SAC Capital Management and Sigma Capital Management. By
                    reason of the provisions of Rule 13d-3 of the Securities
                    Exchange Act of 1934, as amended, each of (i) SAC Capital
                    Advisors, SAC Capital Management and Mr. Cohen may be deemed
                    to own beneficially 38,150 Shares (constituting less than
                    0.1% of the Shares outstanding) and (ii) Sigma Capital
                    Management and Mr. Cohen may be deemed to own beneficially
                    1,471,520 Shares (constituting approximately 5.3% of the
                    Shares outstanding). Each of SAC Capital Advisors, SAC
                    Capital Management, Sigma Capital Management and Mr. Cohen
                    disclaim beneficial ownership of any of the securities
                    covered by this statement.


                                 Page 10 of 13




Item 5              Ownership of Five Percent or Less of a Class:
                    ---------------------------------------------

                    If this statement is being filed to report the fact that as
                    of the date hereof the reporting person has ceased to be the
                    beneficial owner of more than five percent of the class of
                    securities, check the following. [ ]

Item 6              Ownership of More than Five Percent on Behalf of Another
                    --------------------------------------------------------
                    Person:
                    -------

                    Not Applicable

Item 7              Identification and Classification of the Subsidiary Which
                    ---------------------------------------------------------
                    Acquired the Security Being Reported on By the Parent
                    -----------------------------------------------------
                    Holding Company:
                    ----------------

                    Not Applicable

Item 8              Identification and Classification of Members of the Group:
                    ----------------------------------------------------------

                    Not Applicable

Item 9              Notice of Dissolution of Group:
                    -------------------------------

                    Not Applicable

Item 10             Certification:
                    --------------

     By signing below the signatory certifies that, to the best of his knowledge
and belief, the securities referred to above were not acquired and are not held
for the purpose of or with the effect of changing or influencing the control of
the issuer of the securities and were not acquired and are not held in
connection with or as a participant in any transaction having that purpose or
effect.


                                 Page 11 of 13




                                    SIGNATURE
                                    ---------

     After reasonable inquiry and to the best of my knowledge and belief, I
certify that the information set forth in this statement is true, complete and
correct.

Dated: January 7, 2005


S.A.C. CAPITAL ADVISORS, LLC

By: /s/ Peter Nussbaum
    ------------------------------
Name:  Peter Nussbaum
Title: Authorized Person


S.A.C. CAPITAL MANAGEMENT, LLC

By: /s/ Peter Nussbaum
    ------------------------------
Name:  Peter Nussbaum
Title: Authorized Person


S.A.C. MULTIQUANT FUND, LLC

By: /s/ Peter Nussbaum
    ------------------------------
Name:  Peter Nussbaum
Title: Authorized Person


SIGMA CAPITAL MANAGEMENT, LLC

By: /s/ Peter Nussbaum
    ------------------------------
Name:  Peter Nussbaum
Title: Authorized Person


SIGMA CAPITAL ASSOCIATES, LLC

By: /s/ Peter Nussbaum
    ------------------------------
Name:  Peter Nussbaum
Title: Authorized Person


                                 Page 12 of 13




STEVEN A. COHEN

By: /s/ Peter Nussbaum
    ------------------------------
Name:  Peter Nussbaum
Title: Authorized Person


                                 Page 13 of 13