Special values of L-functions for GL₅ × GL₄

Let Π be a cuspidal automorphic representation of GL₅(AQ), and let Σ = Ind(π, χ₂|·|1/2, χ₃|·|-1/2) be an automorphic representation of GL₄(AQ) induced from the standard parabolic subgroup of the form $(2, 1, 1)$ where π is a cuspidal automorphic representation of GL₂(AQ). Assume that Π∞ and Σ∞ 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, Π × Σ). Following Mahnkopf, we prove a result about L(1/2, Π × Σ), and as a corollary, obtain an algebraicity result for the ratio L(1, Π × χ)/L(1, Π × χ'), where χ,χ' are finite order Hecke characters such that χ∞ = χ'∞ = sgn.