Archimedean zeta integrals on $ {\rm SO}_{2n} \times {\rm GL}_n$ and $ {\rm Sp}_{2n} \times {\rm GL}_n $

We evaluate the archimedean Rankin-Selberg zeta integrals associated with spherical principal series representations of ${\rm SO}_{2n}(\mathbf R) \times {\rm GL}_n(\mathbf R) $ and $ {\rm Sp}_{2n}(\mathbf R) \times {\rm GL}_n(\mathbf R) $, where $ {\rm SO}_{2n} $ is split and $n\ge2$. We obtain expl