Definition:Completed Dirichlet L-Function

Definition
Let $\chi$ be a primitive Dirichlet character to the modulus $q \geq 1$.

Let $\displaystyle \kappa = \frac12\left(1 - \chi(-1)\right)$.

Let $\delta = 1$ if $\chi$ is the principal character, and $0$ otherwise.

The completed Dirichlet $L$-function for $\chi$ is defined to be


 * $\displaystyle \Lambda(s,\chi) = \frac{1 + \kappa}{2}\left( s(1-s) \right)^\delta \Gamma\left( \frac{s+\kappa}2 \right)L(s,\chi)$

where $L(s,\chi)$ is the Dirichlet $L$-function for $\chi$.