A question about function related to Riemann zeta function

Question

Let $\zeta(s)$ be a Riemann zeta function. It has only one pole at $s=1$

Consider now function: $G(s)=s(s-1)\zeta(\frac{1}{s})$

I have some questions:

Is $G(s)$ holomorphic everywhere including points $s=1$ and $s=0$?

I wonder whether $G(s)$ is well defined at point $s=0$

If so, then how does Taylor series of $G(s)$ looks like? I am intrested in Taylor series about the center $s=0$

Any help will be appreciated