Eisenstein Cohomology and Critical Values of Certain $L$-Functions: The Case $G_2$

We establish results on the rationality of ratios of successive critical values of Langlands-Shahidi $L$-functions, as they appear in the constant term of the Eisenstein series associated with the exceptional group of type $G_2$ over a totally imaginary number field. Furthermore, we prove the rationality of the critical values for each $L$-function in the products, such as the symmetric cube $L$-functions. Our method generalizes the Harder-Raghuram method to cases where multiple $L$-functions appear in the constant term and involve an exceptional group. Finally, our results on the automorphic version of Deligne's conjecture align with its motivic counterpart, as demonstrated in the recent work of Deligne and Raghuram.