Functional Equation for Dirichlet L-Functions

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

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

Let $\Lambda(s,\chi)$ be the completed $L$-function for $\chi$.

Let $\tau(\chi)$ denote the Gaussian sum.

Then for all $s \in \C$,


 * $\displaystyle \Lambda(s,\chi) = i^{-\kappa}\frac{\tau(\chi)}{\sqrt{q}}\Lambda(1-s,\overline{\chi})$