  We develop the local theory of the generalized doubling method for the
$m$-fold central extension $Sp_{2n}^{(m)}$ of Matsumoto of the symplectic
group. We define local $\gamma$-, $L$- and $\epsilon$-factors for pairs of
genuine representations of $Sp_{2n}^{(m)}\times\widetilde{GL}_k$ and prove
their fundamental properties, in the sense of Shahidi. Here $\widetilde{GL}_k$
is the central extension of $GL_k$ arising in the context of the
Langlands--Shahidi method for covering groups of $Sp_{2n}\times GL_k$. We then
construct the complete $L$-function for cuspidal representations and prove its
global functional equation. Possible applications include classification
results and a Shimura type lift of representations from covering groups to
general linear groups (a global lift is sketched here for $m=2$).
