Definition:Completed Dirichlet L-Function

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

Let $\kappa = \dfrac 1 2 \paren {1 - \map \chi {-1} }$.

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

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

$\map \Lambda {s, \chi} = \dfrac {1 + \kappa} 2 \paren {s \paren {1 - s} }^\delta \map \Gamma {\dfrac {s + \kappa} 2} \map L {s, \chi}$

where $\map L {s, \chi}$ is the Dirichlet $L$-function for $\chi$.