Dynamics of multiples of zeta by exponential functions

We study the family of functions $V_z:s \mapsto ζ(s) \exp (zs)$ with numerical experiments. For selected sequences $z_n=x_nu$ on rays, our experiments suggest that suitable fixed points of $V_{z_n}$ approach nontrivial zeros of $ζ$ in a nearly logarithmic pattern. We formulate this behavior as conje