Special values of L-functions for GL₅ GL₄

Let be a cuspidal automorphic representation of GL₅ (AQ), and let = Ind (, ₂||^1/2, ₃||^-1/2) be an automorphic representation of GL₄ (AQ) induced from the standard parabolic subgroup of the form (2, 1, 1) where is a cuspidal automorphic representation of GL₂ (AQ). Assume that _ and _ are cohomological with respect to the trivial representation in which case s=1/2 is a critical point for the Rankin-Selberg L-function L (s, ). Following Mahnkopf, we prove a result about L (12, ), and as a corollary, obtain an algebraicity result for the ratio L (1, ) /L (1, '), where, ' are finite order Hecke characters such that _ = '_ = sgn.