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By studying an explicit form of the Eichler--Shimura map for ${\rm GL}_3$, wedescribe a precise relation between critical values of the complete$L$-function for the Rankin--Selberg convolution ${\rm GL}_3 \times {\rm GL}_2$and the cohomological cup product of certain rational cohomology classes whichare uniquely determined up to rational scalar multiples from the cuspidalautomorphic representations under consideration. This refines rationalityresults on critical values due to Raghuram et al.

A cohomological interpretation of archimedean zeta integrals for ${\rm GL}_3\times {\rm GL}_2$