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We consider a new integral representation for $L(s_1, \Pi \times \tau_1)L(s_2, \Pi \times \tau_2),$ where $\Pi$ is a globally generic cuspidalrepresentation of $GSp_4,$ and $\tau_1$ and $\tau_2$ are two cuspidalrepresentations of $GL_2$ having the same central character. As andapplication, we find a new period condition for two such $L$ functions to havea pole simultaneously. This points to an intriguing connection between aFourier coefficient of a residual representation on $GSO(12)$ and a thetafunction on $\widetilde{Sp}(16).$ A similar integral on $GSO(18)$ fails tounfold completely, but in a way that provides further evidence of a connection.

A Multi-variable Rankin-Selberg Integral for a Product of $GL_2$-twisted Spinor $L$-functions