Completed Dirichlet L-function ($\Lambda (s,\chi )=(q/\pi {)}^{(s+a)/2}\Gamma ((s+a)/2)L(s,\chi )$)

For a primitive nonprincipal Dirichlet character, the theta Mellin integral makes this an entire function satisfying $\Lambda (s,\chi )={\epsilon }_{\chi }\Lambda (1-s,\overline{\chi })$. The gamma poles cancel the trivial zeros of a Dirichlet L-function. For the principal Dirichlet character whose conductor of a Dirichlet character is one the completion is meromorphic, with poles at zero and one.