The Beurling algebras
$l^1({\cal D},\omega)\;({\cal D}={\bb N},{\bb Z})$
that are semi-simple, with compact Gelfand transform, are considered. The paper gives a necessary and sufficient condition (on
$\omega$
) such that
$l^1({\cal D},\omega)$
possesses a uniform quantitative version of Wiener's theorem in the sense that there exists a function
$\phi:]0,+\infty[\longrightarrow ]0,+\infty[$
such that, for every invertible element
$x$
in the unit ball of
$l^1({\cal D},\omega)$
, we have
\[
\|x^{-1}\|\le \phi(r(x^{-1}))\quad r(x^{-1})\hbox{ is the spectral radius of }x^{-1}.
\]