The critical values of certain Dirichlet series

Abstract

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.