Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$. From the Class number formula and the functional equation of the zeta function, the order of $s=0$ as zero of $\zeta_K$ is $n-1$ and $\zeta$ has simple pole at $s=1$ with residue $1$. So the limit $\lim_{s\to 0} f(s)$ exists and $f(s)$ is holomorphic function on neiberhood of $s=0$.

Is there a way to calculate the derivative $f'(0)$ ?