On the Riemann zeta-function and analytic characteristic functions

Set $f(s) :=1/(\sin(\pi s/4)q(1/2 + s))$ with $q(s) := \pi^{-s/2} \, 2 \Gamma (1 + s/2)(s-1) \zeta (s)$. The Riemann hypothesis, RH, and the simple zeros conjecture, SZC, together with conjectures advanced by the author are used to show that $f(s)$on each vertical strip $V_{4n}$ of $s$ with $4n < {\rm Re} \, (s) < 4(n+1)$ provides an analytic characteristic function, $(-1)^n \cdot f(s) = \int_R (dy) e^{sy} P_{4n} (y)$ with $P_{4n} (y)$ positive. The essential case with $n = 0$ implies RH. A formula is obtained for $P_{4n} (y)$, which for $y$ negative involves the critical zeros. An alternative formula is obtained for $P_{4n} (y)$, without relying on RH, SZC or other unproven conjectures. It does not involve the critical zeros. Analogous resultsfor the cases of the Dirichlet $L$-functions and the Ramanujan tau Dirichlet $L$-function are conjectured.