Local newforms and formal exterior square L-functions

Let F be a non-archimedean local field of characteristic zero. Jacquet and Shalika attached a family of zeta integrals to unitary irreducible generic representations $\pi$ of GL_n(F). In this paper, we show that Jacquet-Shalika integral attains a certain L-function, so called the formal exterior square L-function, when the Whittaker function is associated to a newform for $\pi$. By consideration on the Galois side, formal exterior square L-functions are equal to exterior square L-functions for some principal series representations.