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We give a two variable Rankin-Selberg integral inspired by consideration ofGarrett's pullback formula. For a globally generic cusp form on$\mathrm{GL}_2\times \mathrm{GSp}_4$, the integral represents the product ofthe $\mathrm{Std}\times \mathrm{Spin}$ and $\mathbf{1} \times \mathrm{Std}$$L$-functions. We prove a result concerning an Archimedean principal seriesrepresentation in order to verify a case of Jiang's first-term identityrelating certain non-Siegel Eisenstein series on symplectic groups. Using it,we obtain a new proof of a known result concerning possible poles of these$L$-functions.

A multivariate integral representation on $\mathrm{GL}_2 \times \mathrm{GSp}_4$ inspired by the pullback formula