Dirichlet series
| $L(s,\rho)$ = 1 | − 2-s + 4-s − 8-s − 9-s − 2·13-s + 16-s + 17-s + 18-s + 25-s + 2·26-s − 32-s − 34-s − 36-s − 49-s − 50-s − 2·52-s + 2·53-s + 64-s + 68-s + 72-s + 81-s − 2·89-s + 98-s + 100-s − 2·101-s + 2·104-s − 2·106-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(68\) = \(2^{2} \cdot 17\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 68,\ (0, 1:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,\rho) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Particular Values
\[L(1/2,\rho) \approx 0.3327885867\]
\[L(1,\rho) \approx 0.5583995415\]