According to a [conjecture p.4](http://dml.cz/dmlcz/136881)
$|\zeta(\frac12 -\Delta + it))| > |\zeta(\frac12 + \Delta + i t|$
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)?
> 
> 
> 



Inequality for the modulus of Riemann zeta on horizontal lines and alleged partial result of Maple
For fixed $t>12$, let us consider for $0\leq\sigma\leq \frac{1}{2}$ the function


$$ G(\sigma):=|\pi^{-(\sigma-it)/2}\Gamma(\sigma+it)|^2 = \pi^{-\sigma}|\Gamma(\sigma+it)|^2. $$


Following the accepted answer [here](https://mathoverflow.net/questions/89324/are-all-zeros-of-gammas-pm-gamma1-s-on-a-line-with-real-part-frac12), we see that


$$ \frac{d}{d\sigma}\log G(\sigma)=-\sigma\log\pi + \psi(\sigma+it) + \psi(\sigma-it)$$


$$\geq -\frac{1}{2}\log\pi + 2(1 - \gamma) - 
\sum\_{n=1}^{\infty} \frac{2}{n^2 + t^2} > 0.27-\frac{\pi}{t}>0,$$


whence $G(\sigma)$ is increasing on $[0,\frac{1}{2}]$. 


It follows that $|F(s)|>1$ for $0<\Re(s)<\frac{1}{2}$ and $|\Im s|>24$.


