  We characterize the cuspidal representations of $G_2$ whose standard
$\mathcal{L}$-function admits a pole at $s=2$ as the image of Rallis-Schiffmann
lift for the commuting pair $\left(\widetilde{SL_2}, G_2\right)$ in
$\widetilde{Sp_{14}}$. The image consists of non-tempered representations. The
main tool is the recent construction, by the second author, of a family of
Rankin-Selberg integrals representing the standard $\mathcal{L}$-function.
