Derivatives and exceptional poles of the local exterior square L-function for GL_m

Let π be an irreducible admissible representation of GL_m(F) , where F is a non-archimedean local field of characteristic zero. In 1990’s Jacquet and Shalika established an integral representation for the exterior square L-function. We complete, following the method developed by Cogdell and Piatetski-Shapiro, the computation of the local exterior square L-function L(s, π, ∧ ²) via the integral representation in terms of L-functions of supercuspidal representations by a purely local argument. With this result, we show the equality of the local analytic L-functions L(s, π, ∧ ²) via the integral representation for the irreducible admissible representation π for GL_m(F) and the local arithmetic L-functions L(s, ∧ ²(ϕ(π))) of its Langlands parameter ϕ(π) through local Langlands correspondence.