On Kronecker limit formulas for real quadratic fields Original Research Article

Let image be the partial zeta function attached to a ray class image 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 Zagierʹs 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 Shintaniʹs invariant image, which is related to image, when we change the signature of image.