This paper p.4 claims:

Corollary C. Assume RH. For all large $t$ we have

$$|\zeta(\frac12 +it)| \le \exp\left(\frac38 \frac{\log{t}}{\log{\log{t}}}\right) \qquad (1) $$

$t$ a Gram points often appears counterexample to (1) according to mpmath, pari and maple. E.g. for $t=2381374874120.4, \, 352.4788 \not \le 24.2954 $

Other possible counterexamples $t$ are $$4992394.753, 42653554.76625,3293531640.5520, 29538618461.012969578$$ $$267653395649.1305498,2445999556058.1,2381374874120.4$$

The last one is $13$ digits.

How is "large" defined?

**Added 2013-06-28** According to Computations of the Riemann zeta function the inequality fails for $t \sim 3.925 \cdot 10^{31}$.

For $t=39246764589894309155251169284104.05199$, $\zeta(1/2 + it)=15837.8712 + 3604.9344i, |\zeta(1/2 + it)|=16242.95904$, while the bound from the inequality is $580.2737$ and $\log\log{t}=4.28$.