Published online by Cambridge University Press: 20 November 2018
Inspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\text{GS}{{\text{p}}_{4}}$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\text{PG}{{\text{L}}_{4}}$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group $\text{S}{{\text{L}}_{\text{4}}}\left( \text{C} \right)$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\text{PGU}\left( 2,\,2 \right)$ representing the product of the degree $8$ standard $L$-function and the degree $6$ exterior square $L$-function.