According to a conjecture p.4 $|\zeta(\frac12 -\Delta + it))| > |\zeta(\frac12 + \Delta + i t|$ for $0 < \Delta < \frac12$ and $|t| > 2 \pi +1$. Since $\zeta(\overline{s}) = \overline{\zeta(s)}$, this is equvalent to $|\zeta(s)| > |\zeta(1-s)| $ for $0 < \sigma < \frac12$ and $t$ large enough.

Set $$ F(s) = {\frac {\Gamma \left( 1/2-1/2\,s \right) {\pi }^{-1/2+1/2\,s}{\pi }^{ 1/2\,s}}{\Gamma \left( 1/2\,s \right) }}$$

Then from the functional equation $$\frac{\zeta(s)}{\zeta(1-s)} = F(s) $$ and

$$\frac{|\zeta(s)|}{|\zeta(1-s)|} = |F(s)| $$

$|F(s)| > 1$ for $0 < \sigma < \frac12$ and $t$ large enough would imply the conjecture unless $s$ is a zero of zeta off the critical line.

In Maple 13 using `with(MultiSeries);`

and assuming $0 < \sigma < 1/2$
we get:

$$ \lim_{t \to \infty} |F(\sigma+ it)| = \infty , \, 0 < \sigma < 1/2 $$

Looks like if Maple's result is correct this would mean the conjecture is true at infinity, unless $s$ is a zero off the critical line.

Is Maple's result true?

Proof that $|F(s)| > 1$ for $t$ large enough (except at zeros)?