Sandwiching the Riemann hypothesis

We consider a system of three analytic functions, two of which are known to have all their zeros on the critical line (s) ==1/2. We construct inequalities which constrain the third function, (s), on (s) =0 to lie between the other two functions, in a sandwich structure. We investigate what can be said about the location of zeros and radius of convergence of expansions of (s), with promising results.