Relation between zeta and L-functions, see Kim’s notes. Partial answer (Kim): We don’t have a good canonical def of the complete zeta function. Example for elliptic curve of partial zeta vs partial L-function. In this case, have recipe for bad factors, and also a canonical model, both give same answers. Agreement between etale cohomology of fiber of good model with inertia fixed points of of the thing / Q.
Can we ever hope to prove FE by some form of PD?? What is the analogue in the function field case?
Why not state the conjecture in terms of schemes over Z rather than varieties? We need to assume regular model anyway, right??
nLab page on Beilinson conjectures QUESTIONS