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