Root of the $\zeta(s) = s$

Question

What is the root $s_0$ of the equation $\zeta(s) = s$, where $\zeta(s)$ is Euler zeta function? This point $s_0$ has obvious property: the segment $(1,s_0]$ to the left of it is mapping on the half-line $[s_0, \infty )$ to the right. Is it expressed through Euler–Mascheroni constant $\gamma$?

Answer 1

They are not known to possess a closed form expression. Their numerical values are

$x_+=\quad1.833772651680271396245648589441523592180\ldots \\x_-=-~0.2959050055752139556472378310830480\ldots$

See OEIS A$69995$ and OEIS A$69857$.