Large Z(t) values

Applying the RS-PEAK algorithm the Riemann Zeta Search Project found millions of large values  where Z(t)>1.000.

Define the function \varphi(t)=\frac{\log{|Z(t)|}}{\log(t)}. From the theorem of Bourgain \zeta(1/2+it)= O(t^{13/84+\epsilon}) for every \epsilon>0, and assuming the Riemann hypothesis \lim\limits_{t \rightarrow \infty}{\varphi(t)}=0\, . Finding extremely large Z(t) values with large \varphi(t) is challenging.

The top 37 largest Z(t) with the appropriate \varphi(t) found and verified by the Riemann Zeta Search Project are the followings:

Rendered by