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Euler L-Function

A special case of the Artin L-Function for the Polynomial $x^2+1$. It is given by

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

$\displaystyle \chi^-(p)$ $\textstyle \equiv$ $\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.


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

© 1996-9 Eric W. Weisstein