Laumon introduced the local Fourier transform for $\ell$-adic Galois
representations of local fields, of equal characteristic $p$ different from
$\ell$, as a powerful tool to study the Fourier-Deligne transform of
$\ell$-adic sheaves over the affine line. In this article, we compute
explicitly the local Fourier transform of monomial representations satisfying a
certain ramification condition, and deduce Laumon's formula relating the
epsilon factor to the determinant of the local Fourier transform under the same
condition.