  Let $\Gamma\backslash\mathbb H^3$ be a finite-volume quotient of the
upper-half space, where $\Gamma\subset {\rm SL}(2,\mathbb C)$ is a discrete
subgroup. To a finite dimensional unitary representation $\chi$ of $\Gamma$ one
associates the Selberg zeta function $Z(s;\Gamma;\chi)$. In this paper we prove
the Artin formalism for the Selberg zeta function. Namely, if $\tilde\Gamma$ is
a finite index group extension of $\Gamma$ in ${\rm SL}(2,\mathbb C)$, and
$\pi={\rm Ind}_{\Gamma}^{\tilde\Gamma}\chi$ is the induced representation, then
$Z(s;\Gamma;\chi)=Z(s;\tilde\Gamma;\pi)$. In the second part of the paper we
prove by a direct method the analogous identity for the scattering function,
namely $\phi(s;\Gamma;\chi)=\phi(s;\tilde\Gamma;\pi)$, for an appropriate
normalization of the Eisenstein series.
