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Let \(G=\mathrm{GL}_{2n}\) over a totally real number field F and \(n\ge 2\). Let \(\Pi \) be a cuspidal automorphic representation of \(G(\mathbb {A})\), which is cohomological and a functorial lift from SO\((2n+1)\). The latter condition can be equivalently reformulated that the exterior square L-function of \(\Pi \) has a pole at \(s=1\). In this paper, we prove a rationality result for the residue of the exterior square L-function at \(s=1\) and also for the holomorphic value of the symmetric square L-function at \(s=1\) attached to \(\Pi \). As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gauß sums of Hecke characters.

Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)