  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 cuspidal
representation of $GSp_4,$ and $\tau_1$ and $\tau_2$ are two cuspidal
representations of $GL_2$ having the same central character. As and
application, we find a new period condition for two such $L$ functions to have
a pole simultaneously. This points to an intriguing connection between a
Fourier coefficient of a residual representation on $GSO(12)$ and a theta
function on $\widetilde{Sp}(16).$ A similar integral on $GSO(18)$ fails to
unfold completely, but in a way that provides further evidence of a connection.
