  We introduce a local zeta-function for an irreducible admissible
supercuspidal representation $\pi$ of the metaplectic double cover of $\SL_2$
over a non-archimedean local field of characteristic zero. We prove a
functional equation of the local zeta-functions showing that the gamma factor
is given by a Mellin type transform of the Bessel function of $\pi$. We obtain
an expression of the gamma factor, which shows its entireness on $\C$.
Moreover, we show that, through the local theta-correspondence, the local
zeta-function on the covering group is essentially identified with the local
zeta-integral for spherical functions on ${\rm PGL}_2\cong {\rm SO}_3$
associated with the prehomogenous vector space of binary symmetric matrices.
