  The article describes a purely topological counterpart of the
$\epsilon$-factorization of constants in the functional equations (which is a
key ingredient in the interplay between L-functions and classical automorphic
forms). We consider the determinant of the cohomology of a constructible sheaf
F on a real analytic manifold X (or a bit more precise object, which is
$R\Gamma(X,F)$ seen as a homotopy point of the K-theory spectrum), and show
that it can be "computed" by means of a "spectral" version of the
Dubson-Kashiwara formula, which yields, in particular, the
$\epsilon$-factorization format. This picture may lead to a better
understanding of a recent work of Bloch-Deligne-Esnault on the determinant of
the period matrix.
