Let F a locally compact non-Archimedean field, of residue characteristic p, and ψ a nontrivial additive character of F. Let σ, σ′ be irreducible representations of the absolute Weil group of F, each of degree a power of p and not induced from a nontrivial unramified extension of F. We give a formula for the value at $s=½ of the ϵ-factor ϵ (σ ⊗ σ ',ψ,s)$, modulo roots of unity in ${\Bbb C}$ of order a power of p. Via the Langlands correspondence, we get an analogous formula for supercuspidal representations of ${\rm GL}_n(F)$.