Critical values of L-functions of residual representations of GL4

In this paper, we prove rationality results for critical values for [Formula: see text]-functions attached to representations in the residual spectrum of [Formula: see text]. We use the Jacquet–Langlands correspondence to describe their partial [Formula: see text]-functions via cuspidal automorphic representations of the group [Formula: see text] over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple.