In Theorem 9.12, Titchmarsh (The Theory of the Riemann Zeta Function) proved that

For every large positive T, $\zeta(s)$ has a zero $\beta+i\gamma$ satisfying $$ |\gamma-T|<\frac{A}{\log\log\log T} $$

Is it possible to determine $A$ and $T$ without assuming the Riemann hypothesis?

Or

Are there any other known results (with explicit) around this question?