On Kronecker limit formulas for real quadratic fields

Let ζ (s, C) be the partial zeta function attached to a ray class C of a real quadratic field. We study this zeta function at s = 1 and s = 0, combining some ideas and methods due to Zagier and Shintani. The main results are (1) a generalization of Zagier's formula for the constant term of the Laurent expansion at s = 1, (2) some expressions for the value and the first derivative at s = 0, related to the theory of continued fractions, and (3) a simple description of the behavior of Shintani's invariant X (C), which is related to ζ^′ (0, C), when we change the signature of C.