Functional Equation for Dirichlet L-Functions

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

Let $\map \Lambda {s, \chi}$ be the completed $L$-function for $\chi$.

Let $\map \tau \chi$ denote the Gaussian sum.

Then for all $s \in \C$:


 * $\map \Lambda {s, \chi} = i^{-\kappa} \dfrac {\map \tau \chi} {\sqrt q} \map \Lambda {1 - s, \overline \chi}$

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