On the explicit upper bound of $|\log\zeta(s)|$ near $\Re(s)=1$

Question

I learn from Montgomery & Vaughan's Multiplicative Number Theory I: Classical theory that there exists $c_1,c_2>0$ such that whenever $\sigma\ge1-c_1/\log|t|,|t|\ge4$ there is $$ |\log\zeta(\sigma+it)|\le\log\log|t|+c_2,\tag1 $$ As I want to bound $|\zeta(\sigma+it)|^k$ uniformly in $k$, I wonder whether there are literature that provides a version of (1) that has $c_1$ and $c_2$ explicitly calculated.

Answer 1

Theorem 1 here shows that if $|t|\geq 3$ and $\frac{1}{2}\leq\sigma\leq 1$, then $|\zeta(\sigma+it)|\leq 76.2|t|^{4.45(1-\sigma)^{3/2}}(\log |t|)^{2/3}$. This is the best known explicit bound for $|\zeta(\sigma+it)|$ when $\sigma\geq 1-e^{-1}(\log|t|)^{-2/3}(\log\log|t|)^{-1/3}$.