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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 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 Sp(16). A similar integral on GSO(18) fails to unfold 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