Let $\chi(s)=2^s \pi^{s-1}\sin(\pi s/2) \Gamma(s)$ Gamma(1-s)$so that$\zeta(s)=\chi(s)\zeta(1-s)$. You are asking about the curve$|\chi(s)|=1$. As you have observed,$|\chi(1/2+it)|=1$for real$t$. There is a partial converse to this statement, namely that there is a positive absolute constant$C_0$such that if$|\chi(\sigma+it)| = 1$with$0 \le \sigma \le 1$and$|t| \ge C_0$, then$\sigma=1/2$. A simple proof can be found in Lemma 6.1 of S. M. Gonek "Finite Euler products and the Riemann hypothesis" Trans. Amer. Math. Soc. 364 (2012), 2157-2191. This paper is also on the arXiv. Gonek states that$C_0<6.3$so it seems that phenomena in your pictures stops shortly after the ranges you plotted. 1 Let$\chi(s)=2^s \pi^{s-1}\sin(\pi s/2) \Gamma(s)$so that$\zeta(s)=\chi(s)\zeta(1-s)$. You are asking about the curve$|\chi(s)|=1$. As you have observed,$|\chi(1/2+it)|=1$for real$t$. There is a partial converse to this statement, namely that there is a positive absolute constant$C_0$such that if$|\chi(\sigma+it)| = 1$with$0 \le \sigma \le 1$and$|t| \ge C_0$, then$\sigma=1/2$. A simple proof can be found in Lemma 6.1 of S. M. Gonek "Finite Euler products and the Riemann hypothesis" Trans. Amer. Math. Soc. 364 (2012), 2157-2191. This paper is also on the arXiv. Gonek states that$C_0<6.3\$ so it seems that phenomena in your pictures stops shortly after the ranges you plotted.