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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?

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Theorem 9.12 in Titchmarsh says (in his shorthand style) there exists a constant $A$ such that for all sufficiently large T, (etc.)

The proof uses the Borel-Caratheodory theorem, and can be made effective if you really really want it. Titchmarsh has a series of seven successive constants $A_1, A_2,\ldots A_6, A$ with the final $A$ being the constant you reference above. This is not conditional on the Riemann Hypothesis.

It's not clear how your actual question relates to your title.

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do you want to determine T also? I thought the result should hold for all T?

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@Dierk. If so, it would be better, if not just to know if it is not large. for example if $A=1$ and $T<10^{10}$, it's OK. – asd Jun 12 '12 at 15:08

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