Other functional equations for $\zeta(s)$? 
For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ and $\zeta(s-1)$?

EDIT: My main motivation behind asking this question is I have found such an equation, but I do not know whether such an equation exists in literature. Also, I do not want to appear as if I am promoting my formula here, but rather I am more interested in the works that have been done in such directions.
As per @lhf's request here is my formula, for $\Re(s) > 1$
$$ \zeta(s) + \frac{2}{s-1}\zeta(s-1) = \frac{s}{s-2} - s\int_1^\infty \frac{\{x\}^2}{x^{s+1}} dx$$ where $\{x\}$ is the fractional part of x.

Yes, $\zeta(\overline{s})=\overline{\zeta(s)}$ for $s \in \mathbb C-\{1\}$.
