  We construct variants of the Riemann zeta function with convenient properties
and make conjectures about their dynamics; some of the conjectures are based on
an analogy with the dynamical system of zeta. More specifically, we study the
family of functions $V_z: s \mapsto \zeta(s) \exp (zs)$. We observe convergence
of $V_z$ fixed points along nearly logarithmic spirals with initial points at
zeta fixed points and centered upon Riemann zeros. We can approximate these
spirals numerically, so they might afford a means to study the geometry of the
relationship of zeta fixed points to Riemann zeros.
