I find some explicit expression of derivative of logarithm of Riemann Zeta function $\frac{\zeta'(s)}{\zeta(s)}$ is already found. For example, in this answer link.
I am curious about whether there is some explicit expression for $$\frac{\zeta(s)}{\zeta'(s+1)}$$ The only thing I can think of is to use the respective expression of the $\zeta(s)$ and $\zeta'(s+1)$ and derive something from them. But I think it may not the great approach.
Or is it possible to just get the Order of $\frac{\zeta(s)}{\zeta'(s+1)}$? Thank you for any advice!