Analytic continuation and functional equation of $L(s,\tau)$

There is an entire function $L(s)$ on $\mathbb C$ which agrees with the Dirichlet series $\sum_{n\ge1}\tau(n)n^{-s}$ for $\operatorname{Re}s>13/2$, i.e. the analytic continuation of $L(s,\tau)$, and which satisfies

for every $s\in\mathbb C$ such that neither $s$ nor $12-s$ is a non-positive integer (the poles of $\Gamma$).

This is the functional equation $s\leftrightarrow 12-s$ of the Ramanujan $L$-function.

Formalization Note $\tau$ is RamanujanPetersson.tau, the $n$-th coefficient of Mathlib's $q$-expansion (period $1$) of ModularForm.discriminant, valued in $\mathbb C$. Mathlib's Complex.Gamma takes the value $0$ at the poles of $\Gamma$. The excluded points are exactly those where one side involves such a junk value.