Q2: Cramér's paper is from 1919, a modern and more extensive treatment is in On Cramér's theorem for general Euler products with functional equation (1993).

Q1: There is a functional relation for $V(z)$, but no "explicit expression" I know of.

Q2: Cramér's paper is from 1919, a modern and more extensive treatment is in On Cramér's theorem for general Euler products with functional equation (1993).