Let $K=\mathbb{Q}(i)$ be the field of Gaussian rationals. Let $\zeta_K(s)$ be the Dedekind zeta function associated to K, defined by $$\zeta_K(s):=\sum_{\mathfrak{a}\subseteq \mathbb{Z}[i]}\frac{1}{\mathrm{N}(\mathfrak{a})^s},$$ where $\mathfrak{a}$ ranges over all the nonzero ideals of $\mathbb{Z}[i]=\mathcal{O}_K$. It is a well-known fact that $\zeta_K(s)$ can be extended to a meromorphic function on $\mathbb{C}$ that has a simple pole at $s=1$ with residue $\pi/4$ and no other poles.
My question is that:
Can the following (finite) limit $$\lim_{s\to 1}\left(\zeta_K(s)-\dfrac{\pi}{4(s-1)}\right)$$ be explicitly computed (or at least in terms of the other constants, like the Euler-Mascheroni constant $\gamma$, etc)?
From the factorization
$$\zeta_K(s)=\zeta(s)L(s,\chi)$$
where $\chi$ is the unique nontrivial character mod $4$, and the expansions
$$\zeta(s)=\frac1{s-1}+\gamma+\cdots$$
and
$$L(s,\chi)=L(1,\chi)+(s-1)L'(1,\chi)+\cdots$$
it follows that
$$\zeta_K(s)=\frac{L(1,\chi)}{s-1}+\gamma L(1,\chi)+L'(1,\chi)+\cdots$$
where $L(1,\chi)=\pi/4$ and
$$L'(1,\chi)=-\sum_{n=1}^\infty\frac{\chi(n)\log n}n.$$
This series can be computed explicitly using Kummer's formula for $\log \Gamma$.