Zero-crossings along the critical line s = 1/2 + it via Hardy's Z(t) function
Along the critical line $\text{Re}(s) = 1/2$, the Riemann zeta function can be evaluated using Hardy's $Z(t)$ function, defined as $Z(t) = e^{i \theta(t)} \zeta(1/2 + it)$, where $\theta(t)$ is the Riemann-Siegel theta function. $Z(t)$ is real-valued for real $t$, and its zeros correspond exactly to the non-trivial zeros of $\zeta(s)$.
Zero-crossings of $Z(t)$ mark points where $|\zeta(1/2 + it)| = 0$. The Gram points $g_n$ (where $\theta(g_n) = n\pi$) provide reference milestones along the critical strip, governing Gram's law which predicts that $Z(t)$ alternates sign between consecutive Gram points.
This interactive graph plots $Z(t)$ and $|\zeta(1/2 + it)|$ along the critical axis. Adjusting $t_{\text{center}}$ allows real-time exploration of zero spacing dynamics, Gram point locations, and amplitude fluctuations across higher energy bands.