I am studying de la Vallee Poussin's proof of the zero-free region of the Riemann zeta function. Assuming the following bound
$$\Re \frac{\Gamma'}{\Gamma}(s/2) \leq \log \vert s/2\vert + \min \Big(\frac{\pi/2}{\Im s}, \frac{1}{\Re s}\Big),$$ what is the best constant $C$ that can be obtained in the zero-free region
$$\Re s > 1 - \frac{1}{C\log \vert \Im s \vert}, \quad\vert \Im s\vert > 2\,? $$
So, basically, I am trying to make the proof given by Poussin explicit. The main place where I am stumped is how do I use the above bound, though I can see how the bound follows from the Stirling's asymptotic formula for Gamma. Any ideas and/or hints? Thanks in advance....