Dirichlet series
| $\zeta_K(s)$ = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s + 2·11-s + 2·12-s + 16-s + 2·17-s + 3·18-s + 2·19-s + 2·22-s + 2·24-s + 25-s + 4·27-s + 32-s + 4·33-s + 2·34-s + 3·36-s + 2·38-s + 2·41-s + 2·43-s + 2·44-s + 2·48-s + 49-s + 50-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda_K(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr
=\mathstrut & \, \Lambda_K(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(8\) = \(2^{3}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 8,\ (\ :0),\ 1)$ |
Euler product
\[\begin{aligned}
\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{8}(3, \cdot))\)
Particular Values
\[\zeta_K(1/2) \approx -1.607005366\]
Pole at \(s=1\)