  We construct the Weil-\'etale cohomology and Euler characteristics for a
subclass of the class of $\mathbb{Z}$-constructible sheaves on the spectrum of
the ring of integers of a totally imaginary number field. Then we show that the
special value of an Artin L-function of toric type at zero is given by the
Weil-\'etale Euler characteristic of an appropriate $\mathbb{Z}$-constructible
sheaf up to signs. As applications of our result, we will derive some classical
formulas of special values of L-functions of tori and their class numbers.
