Answer by Henri Cohen for Do the roots of this equation involving two Euler products all reside on the critical line?

This problem (and more general ones) was solved completely andunconditionally (i.e., without any RH) by P.R. Taylorbefore the second world war, but I don't remember the reference,it should be easy to Google. Download for example the paper byMcPhedran and Poulton "The Riemann Hypothesis for Symmetrised combinations of the Zeta Function"