Calcul d'une valeur d'un facteur epsilon par une formule int\'egrale

Let d and m be two natural numbers of distinct parities. Let $\pi$ be an admissible irreducible tempered representation of GL(d,F), where F is a p-adic field. We assume that $\pi$ is self-dual. Then we can extend $\pi$ as a representation $\tilde{\pi }$ of a non-connected group $GL(d,F)\rtimes {1,\theta}$\mathrm{.}Let$\rho$ be a representation of GL(m,F). We assume that it has similar properties as $\pi$. Jacquet, Piatetski-Shapiro and Shalika have defined the factor $ϵ(s,\pi \times \rho ,\psi )$. We prove that we can compute $ϵ(1\mathrm{/}2,\pi \times \rho ,\psi )$ by an integral formula where occur the characters of $\tilde{\pi }$ and $\tilde{\rho }$. It’s similar to the formula which, for special orthogonal groups, computes the multiplicities appearing in the local Gross-Prasad conjecture.