#### DOCUMENTA MATHEMATICA, Vol. 13 (2008), 775-794

Goro Shimura

The Critical Values of Certain Dirichlet Series

We investigate the values of several types of Dirichlet series $D(s)$ for certain integer values of $s,$ and give explicit formulas for the value $D(s)$ in many cases. The easiest types of $D$ are Dirichlet $L$-functions and their variations; a somewhat more complex case involves elliptic functions. There is one new type that includes $\sum_{n=1}^{\infty} (n^2+1)^{-s}$ for which such values have not been studied previously.

2000 Mathematics Subject Classification: 11B68, 11M06, 30B50, 33E05.

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