Going through what it says on page 95 of [Titchmarsh's book on the Zeta function][1], and using his remark about $\mu$ at the bottom of that page, I conclude that there is some constant $C>0$ so that: $$\left|\zeta\left(it\right)\right|\leq C\sqrt{\left|t\right|},\textrm{ }\forall t\in\mathbb{R}$$

In my current work, I need to know an explicit value for this $C$. Graphing the quotient: $$\frac{\left|\zeta\left(2^{t}i\right)\right|}{\sqrt{2^{t}}}$$ gives an apparent global maximum of slightly less than 1.37, occurring at $t≈8.9082$   

Is this rigorously justifiable? I.e., is:

$$\left|\zeta\left(it\right)\right|\leq 1.37\sqrt{\left|t\right|},\textrm{ }\forall t\in\mathbb{R}$$

true? More generally, what's the most accurate estimate of this type which is currently known (with explicit constants)?

  [1]: https://zulfahmed.files.wordpress.com/2018/08/titchmarsh-and-d-r-heath-brown-the-theory-of-the-riemann-zeta-function.pdf