  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 $GL(d,F)\rtimes
\{1,\theta\}$. 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 $\epsilon(s,\pi\times\rho,\psi)$. We prove that we can
compute $\epsilon(1/2,\pi\times\rho,\psi)$ 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.
