  We give two global integrals that unfold to a non-unique model and represent
the partial Spin $L$-function on $\mathrm{GSp}_6$. We deduce that for a wide
class of cuspidal automorphic representations $\pi,$ the partial Spin
$L$-function is holomorphic except for a possible simple pole at $s=1$, and
that the presence of such a pole indicates that $\pi$ is an exceptional theta
lift from $\mathrm{G}_2$. These results utilize and extend previous work of Gan
and Gurevich, who introduced one of the global integrals and proved these facts
for a special subclass of these $\pi$ upon which the aforementioned model
becomes unique. The other integral can be regarded as a higher rank analogue of
the integral of Kohnen-Skoruppa on $\mathrm{GSp}_4$.
