The functions $\xi(s)$ and
$$f(s):=\xi\left(\frac{s-i}{2}\right) + \xi\left(\frac{i+s}{2}\right)$$
grow faster than exponentially on the positive axis, hence they do not satisfy the first bound. This follows from Stirling's approximation for $\Gamma(s)$ and the fact that $\zeta(s)$ tends to $1$ as $\Re s\to\infty$.