It was conjectured by Deligne and Clozel that the algebraicity of critical values of automorphic L-functions for cohomological cuspidal automorphic representations should be expressed in terms of the conjectural motivic periods of the associated motive. Very little is known for the general cases by far. A special class of examples are the symmetric power L-functions of cohomological cuspidal automorphic representations of GL₂ over totally real number fields. In these cases, the conjectural periods are product of the Petersson norms of Hilbert cusp newform and Deligne’s motivic periods for GL₂.

In this talk, we introduce our recent work on the symmetric fourth L-functions. We extend the result of Morimoto based on generalization and refinement of the results of Grobner and Lin to cohomological irreducible essentially conjugate self-dual cuspidal automorphic representations of GL₂ and GL₃ over CM-fields.