Let $\pi$ be an automorphic form on $GL(3,\mathbb A_\mathbb Q)$. 
Do we know any case that
$$\int_0^{T} \left|L(\frac 1 2 + it, \pi)\right| dt \gg T$$ 
holds unconditionally?

I know the conjectured asymptotic formula is $T \log^* T$.