Tamagawa number formula with coefficients over varieties in positive characteristic

We express the order of the pole and the leading coefficient of the L-function of a (large class of) -adic coefficients (any prime) over a quasi-projective variety over a finite field of characteristic p. This is a generalization of the result of Milne-Ramachandran with coefficients. The new key ingredient is the use of F-gauges and their equivalence in the derived category with Raynaud modules proved by Ekedahl.