Computational estimation of the order of $\zeta (\frac{1}{2}+it)$

The paper describes a search for increasingly large extrema (ILE) of $\left\vert \zeta (\frac{1}{2}+it)\right\vert$ in the range $0\leq t\leq 10^{13}$. For $t\leq 10^{6}$, the complete set of ILE (57 of them) was determined. In total, 162 ILE were found, and they suggest that $\zeta (\frac{1}{2} +it)=\Omega (t^{2/\sqrt{\log t\,\log \log t}})$. There are several regular patterns in the location of ILE, and arguments for these regularities are presented. The paper concludes with a discussion of prospects for further computational progress.