Now if $\pi_p$ and $\pi_p'$ are both unramified and $\varphi_p$ and $\varphi_p'$ are both chosen to be spherical vectors (and normalised appropriately), then this local Eulerian integral is equal to $L(s, \pi_p \otimes \pi_p')$, the local component of the Rankin-Selberg $L$-function $L(s,\pi \otimes \pi')$ associated to $\pi$ and $\pi'$. So if both $\pi$ and $\pi'$ are unramified everywhere, then the global Eulerian integral is equal to
\[\prod_{p \leq \infty}L(s, \pi_p \otimes \pi_p') = \Lambda(s, \pi \otimes \pi'),\]
the completed Rankin-Selberg $L$-function (i.e. the finite part $L(s,\pi \otimes \pi) = \prod_p L(s, \pi_p \otimes \pi_p')$ multiplied by the infinite part, which is a product of gamma factors). In particular, this is the case if $\varphi$ is the adèlic lift of an even Hecke-Maaß cusp form $F$ of level $1$ on $\mathrm{GL}_3$ and $\varphi'$ is the adèlic lift of an even Hecke-Maaß cusp form $u$ of level $1$ on $\mathrm{GL}_2$.
Very recently, Hirano, Ishii, and Miyazaki [have announced][1] that they have discovered an explicit description of a pair of vectors satisfying this. The description is unfortunately rather complicated (it involves the decomposition of $\pi_{\infty}$ and $\pi_{\infty'}$ into $K$-types, and choosing explicit vectors in particular $K$-types). Moreover, they don't explicitly state the relation between their vectors and the "natural" choice of vectors; here the "natural" vector is the spherical vector for $\pi_{\infty}$ and the unique vector (up to scalar) lying in the minimal $K$-type of $\pi_{\infty'} = \mathrm{sgn} |\cdot|^{it} \boxplus \mathrm{sgn} |\cdot|^{-it}$, which is the determinant representation of $K = \mathrm{O}(2)$ (which is one-dimensional). Finally, their result must be taken with a pinch of salt, since the proofs are yet to appear, and it builds on work (at least in the complex case) that has some major, as yet uncorrected, errors.
