Algebraic cycles and residues of degree eight $L$-functions of $\mathrm{G}\mathrm{S}\mathrm{p}(4)\times \mathrm{G}\mathrm{L}(2)$

We prove a cohomological formula for non-critical residues of degree eight automorphic $L$-functions of $\mathrm{G}\mathrm{S}\mathrm{p}(4)\times \mathrm{G}\mathrm{L}(2)$ in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky’s integral representation of the $L$-functions.