Denote by $\zeta$ the Riemann zeta function. Since $\zeta(1/2 + it)\neq 0$ for $|t|\leq 14$, it seems to follow that $|\zeta(1/2 + it)|$ is either strictly decreasing or strictly increasing on $(-14,14)$. But what exactly is the behaviour of $|\zeta(1/2 +it)|$ on this region ? I mean, is it strictly increasing or is it strictly decreasing ?
A graphical sketch might help.
It doesn't follow that $\lvert \zeta(1/2 + it) \rvert$ is strictly increasing or decreasing.
In fact, since $\lvert \zeta(1/2 + it)\rvert = \lvert \zeta(1/2 - it) \rvert$ from the functional equation, one cannot have that $\lvert \zeta(1/2 + it) \rvert$ is strictly increasing or decreasing at $0$, since the function is symmetric there.
One can plot this readily using something like sage, or something online like wolfram alpha. You get the following plot.