  We consider the Ihara zeta function $\zeta(u,X//G)$ and Artin-Ihara
$L$-function of the quotient graph of groups $X//G$, where $G$ is a group
acting on a finite graph $X$ with trivial edge stabilizers. We determine the
relationship between the primes of $X$ and $X//G$ and show that $X\to X//G$ can
be naturally viewed as an unramified Galois covering of graphs of groups. We
show that the $L$-function of $X//G$ evaluated at the regular representation is
equal to $\zeta(u,X)$ and that $\zeta(u,X//G)$ divides $\zeta(u,X)$. We derive
two-term and three-term determinant formulas for the zeta and $L$-functions,
and compute several examples of $L$-functions of edge-free quotients of the
tetrahedron graph $K_4$.
