It is known that for certain particular entire functions $f(s)$ of first order, in the circle $|s| = p$, if $\epsilon$ is a positive number as small as desired, the following bound holds:

$$|f(s)| = O(e^{\epsilon p})$$

A classical entire function of first order is Riemann's xi function, written as:

$$\xi(s) = \frac{1}{2} s(s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s)$$

Is it easy to show that Riemann's hypothesis(RH), or Lindeloph's hypothesis(LH), imply this bound? If this bound is assumed, would both these hypothesis follow? It is known that RH implies LH, but whether LH implies RH is still a deep unknown. Consider the function $f(\xi(s)) = f(\xi)$ defined by

$$f(\xi) - \xi\left(\frac{s-i}{2}\right) - \xi\left(\frac{i+s}{2}\right) = 0$$

An interesting question is whether, in the circle $|s| = p$, the mentioned bound $|f(\xi)| = O(e^{\epsilon p})$ holds. The function $f(\xi)$ is interesting in that it is easy to show that all its zeros are real.