  In this paper we prove rationality results of critical values for
$L$-functions attached to representations in the residual spectrum of
$\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to
describe their partial $L$-functions via cuspidal automorphic representations
of the group $\mathrm{GL}_2'(\mathbb{A})$ 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.
