Extreme values of automorphic $L$-functions

We treat $\Omega$-theorems for some automorphic $L$-functons, and for the Rankin–Selberg $L$-function $L(s,f\times f)$ in particular.
For example, as $t$ tends to infinity,
$$ \log\Bigg|L\Biggl(\frac12+it,f\times f\Biggr)\Bigg|=\Omega_+\Biggl(\Biggl(\frac{\log t}{\log\log t}\Biggr)^{1/2}\Biggr), $$
and
$$ \log\big|L(\sigma_0+it,f\times f)\big|=\Omega_+\Biggl(\Biggl(\frac{\log t}{\log\log t}\Biggr)^{1-\sigma_0}\Biggr) $$
for fixed $\sigma_0\in\big(\frac12,1\big)$.