Archimedean zeta integrals for $GL(3)\times GL(2)$

In this article, we give explicit formulas of archimedean Whittaker functions on $GL(3)$ and $GL(2)$. Moreover, we apply those to the calculation of archimedean zeta integrals for $GL(3)\times GL(2)$, and show that the zeta integral for appropriate Whittaker functions is equal to the associated $L$-factors.