For an arbitrary complex number a≠ 0 we consider the distribution of values of the Riemann zeta-function ζ at the a-points of the function Δ which appears in the functional equation ζ(s) = Δ (s) ζ(1 - s). These a-points δa are clustered around the critical line 1 / 2 + iR which happens to be a Julia line for the essential singularity of ζ at infinity. We observe a remarkable average behaviour for the sequence of values ζ(δa).
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