$\zeta(1 + 2/x)$ has a strange, nearly linear behaviour

Question

I was messing around with some infinite sums in $\ell^p$ spaces and I encountered a strange result:

$\zeta\left(1 + \frac{2}{x}\right)$ looks like it is linear in $x$ for $x > 1$! A simple linear regression gives me $\zeta\left(1 + \frac{2}{x}\right)\approx 0.593413 + 0.499801 x$. The greatest difference appears to be relatively small.

Is this true? If so, how could I show this? And if not, why does it appear to be nearly linear?