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