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We develop the local theory of the generalized doubling method for the$m$-fold central extension $Sp_{2n}^{(m)}$ of Matsumoto of the symplecticgroup. We define local $\gamma$-, $L$- and $\epsilon$-factors for pairs ofgenuine representations of $Sp_{2n}^{(m)}\times\widetilde{GL}_k$ and provetheir fundamental properties, in the sense of Shahidi. Here $\widetilde{GL}_k$is the central extension of $GL_k$ arising in the context of theLanglands--Shahidi method for covering groups of $Sp_{2n}\times GL_k$. We thenconstruct the complete $L$-function for cuspidal representations and prove itsglobal functional equation. Possible applications include classificationresults and a Shimura type lift of representations from covering groups togeneral linear groups (a global lift is sketched here for $m=2$).

Doubling Constructions: the complete L-function for coverings of the symplectic group