Using the modular forms database ([1], [2]) I could find the coefficients of the zeta function corresponding to the number field $F = \mathbb{Q}(\sqrt{-23}) \simeq \mathbb{Q}(x)/(x^2 - x +6)$. We could in a literal-minded way write down the terms: $$ \zeta_F(-1) = 1 + 2\cdot 2^0 + 2\cdot 3^0 + 3 \cdot 4^0 + 0 + 4\cdot 6^0 + 4 \cdot 8^0 + 3\cdot 9^0 + 0 + 0 + 6\cdot 12^0 + \dots $$ I am writing a bunch of terms which are basically $n^0=1$ over and over. Number theorists likely read this one as $n^\epsilon$ with $\epsilon \approx 0$ when they write a meromorphic continuation. In the case, $F = \mathbb{Q}$ we just have $\zeta(0) = 1 + 1 +1 + \dots = \sum 1 = -\frac{1}{2}$ as done on Wikipedia.
By the way, could someone remind me why the coefficients jump around so much? It looks like we are solving $p = a^2 + 23b^2$ for the various numbers. Except that: $$ 4 = a^2 + 23 \, b^2 $$ Doesn't have any solutions, so I don't know why the coefficient is $3 \cdot 4^0$. We do have the factorization: \begin{eqnarray*} \zeta_F(s) = \zeta(s) \cdot L(s, \chi_{23}(22, \cdot) ) &=& \prod_p \left( 1 - \frac{1}{p^s} \right)^{-1} \times \prod_p \left( 1 - \frac{\chi_{23}(p)}{p^s} \right)^{-1} \\ &\stackrel{s=0}{=}& \left( \left(1 - \frac{1}{2^0} \right) \times \left(1 - \frac{1}{3^0} \right) \times \left(1 - \frac{1}{5^0} \right) \times \left(1 - \frac{1}{7^0} \right) \times \dots \right) \times \left( \left(1 - \frac{1}{2^0} \right) \times \left(1 - \frac{1}{3^0} \right) \times \left(1 + \frac{1}{5^0} \right) \times \left(1 + \frac{1}{7^0} \right) \times \dots \right) \end{eqnarray*} where $\chi_{23}: \mathbb{Z}^\times \to \{ 1, -1\}$ is an appropriate sequence of signs ([3]) .
There's standard references on Dedekind zeta function $\zeta_F(s)$ and the Birch-Tate conjecture does mentio $\zeta_F(-1)$ the discussion looks very technical. With a computer program it's possible to find the coefficients on the zeta function (and it would be great to see different programs). By the modern standard there's lots of error here, this is maybe closer to what Euler had in mind.
I took the coefficients of the Dedekind zeta function from a list. It would be nice to see a computer program or at least a Sage or PARI script. In fact, $\mathbb{Q}(\sqrt{-23})$ as class number of $3$.
- $a^2 + 23b^2$
- $2a^2 - ab + b^2$
- one more