Exceptional poles of archimedean Rankin-Selberg L-functions for principal series representations of GL(n,R)

We prove that for any pair of irreducible principal series representations $({\pi }_1,{\pi }_2)$ of ${\mathrm{GL}}_n(\mathbb{R})$ in general position, the notions of exceptional pole of type 1 and type 2 coincide. Using this identification, we express the Rankin–Selberg $L$-function $L(s,{\pi }_1\times {\pi }_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of the derivatives of ${\pi }_1$ and ${\pi }_2$.