Holmstrom Zeta values theory

Scholbach explained to us at some point the homotopy invariance of the cohomology theories involved and how taking the product with A1 changes the L-functions but in a very predictable way, some shift and something else I think. Some scratch notes which might be wrong: Motives with compact support (and hence zeta functions) are not homotopy invariant, but $M_c(X \times A1) \simeq M_c(X) \otimes \mathbf{1}(1)[2]$ where the shift gives the reciprocal L-function, and the twist gives a shift in the argument. Hence the zeta function simply becomes shifted one step when taking product with the affine line.

Lichtenbaum talk in Oberwolfach July 2009. Ref to Chinburg et al: Epsilon factors and Arakelov Euler char, for zeta functions and more. Math Res Lett 7 (2000). Lichtenbaum also had doubts about Bloch-Kato formulation in general, for example, no compatibility with FE has been proven, he said.

nLab page on Zeta values theory