Level curves for Zhang's Eta Function

Study of the level curve for the real part of $\eta (s)=0$ with $\eta (s)={\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(s\mathrm{/}2){\zeta }^{\mathrm{\prime }}(s)$ gives a new classification of the zeros of $\zeta (s)$ and of ${\zeta }^{\mathrm{\prime }}(s)$. We conjecture that for type 2 zeros, $\mathrm{lim\hspace{0.17em}inf}({\beta }^{\mathrm{\prime }}-1\mathrm{/}2)\log {\gamma }^{\mathrm{\prime }}=0$ if and only if $\mathrm{lim\hspace{0.17em}inf}({\gamma }^{+}-{\gamma }^{-})\log {\gamma }^{\mathrm{\prime }}=0$, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify ${10}^6$ zeros of ${\zeta }^{\mathrm{\prime }}(s)$ near $T={10}^{10}$. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices.