Every Voevodsky motive over a global field now has a canonical L-function that is multiplicative on exact triangles, with a functional equation in characteristic p.
Kahn Zeta functions and motives , Pure Appl
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Zeta and L functions of Voevodsky motives
Every Voevodsky motive over a global field now has a canonical L-function that is multiplicative on exact triangles, with a functional equation in characteristic p.