Value-Distribution of the Riemann Zeta-Function along its Julia Lines

For an arbitrary complex number $a\neq 0$ we consider the distribution of values of the Riemann zeta-function $\zeta$ at the $a$-points of the function $\Delta$ which appears in the functional equation $\zeta(s)=\Delta(s)\zeta(1-s)$. These $a$-points $\delta_a$ are clustered around the critical line $1/2+i\mathbb{R}$ which happens to be a Julia line for the essential singularity of $\zeta$ at infinity. We observe a remarkable average behaviour for the sequence of values $\zeta(\delta_a)$.