Euler L-Function

A special case of the Artin L-Function for the Polynomial . It is given by

$\begin{displaymath} L(s)=\prod_{p{\rm\ odd\ prime}} {1\over 1-\chi^-(p) p^{-s}}, \end{displaymath}$

where

$\displaystyle \chi^-(p)$	$\textstyle \equiv$	$\displaystyle \left\{\begin{array}{ll} 1 & \mbox{for $p\equiv 1\ \left({{\rm mo... ... -1 & \mbox{for $p\equiv 3\ \left({{\rm mod\ } {4}}\right)$}\end{array}\right.$
	$\textstyle =$	$\displaystyle \left({-1\over p}\right),$

where $\left({-1/p}\right)$ is a Legendre Symbol.

References

Knapp, A. W. ``Group Representations and Harmonic Analysis, Part II.'' Not. Amer. Math. Soc. 43, 537-549, 1996.