  The purpose of this paper is to give a formula for the leading coefficient at
$s=1$ of the $L$-function of one-motives over function fields in terms of
Weil-\'etale cohomology, generalizing the Weil-\'etale version of the Birch and
Swinnerton-Dyer conjecture in the authors' previous work. As a consequence we
express the Tamagawa number of a torus introduced by Ono-Oesterl\'e in terms of
Weil-\'etale cohomology, and reprove their Tamagawa number formula.
