Enquiry on bounds for $\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1.$

Question

Let $\zeta$ be the Riemann zeta function and $n$ be a positive integer. What are the known (conditional and unconditional) bounds for $f(n)=\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1$ ?

By applying the Taylor series for $(s-1)\zeta(s)$ around $s=1$, it seems that one naively obtains $f(n)=O(n^2)$. Is this true ? And are there sharper results ?