We consider here the archimedean zeta integrals for $\mathit{GL}(3)\times \mathit{GL}(2)$ and show that the zeta integral for appropriate Whittaker functions is equal to the associated $L$-factor.

In this article we study \textit{simple vertex operator algebras} whose spaces of characters are 3-dimensional, and satisfy \textit{3rd order (modular) linear differential equations}. We classify such vertex operator algebras with central charge 1/2 or $-68/7$. One of the main results is that these vertex operator algebras have \textit{conformal weights} $\{0,1/2,1/16\}$ or $\{0,-2/7,-3/7\}$, respectively, and are isomorphic to the minimal models of central charge $c=c_{3,4}=1/2$ and $c_{2,7}=-68/7$. Moreover, we give the explicit formulas representing characters of the minimal models with $c=1/2$, $-68/7$ by using the classical Weber functions.