Q(z) = ,Q?(&2 of an integral weight > 0; k is a positive integer; * is an embedding of K into C; r is an element of K0 such that Id2 is its only positive conjugate; 4D-b(* + Ap) + ,cpgp, where b and c, are non-negative integers, p is the complex conjugation, and {qp} is the set of all embeddings of K into C other than * and Ap. It will be shown that the series is convergent for sufficiently large Re (s) and can be continued to a meromorphic function on the whole plane. Now one of our main results will assert that the values ?D(4a) for certain integers , are algebraic numbers times wkPK(*i *)2k I, P (Tv, ,)-2c. where PK(q', A) is a complex number depending only on q and A. We shall actually prove such an algebraicity for the series defined in a similar way with a Hilbert modular form in place of Q (Theorem 9.2). If [K: QI = 2, 9) is essentially of the type