Dirichlet series
| $L(s,\rho)$ = 1 | − 5-s + 8-s − 13-s − 17-s − 23-s + 2·25-s + 27-s − 31-s − 37-s − 40-s − 53-s − 59-s − 61-s + 64-s + 65-s + 85-s − 104-s + 115-s − 2·125-s − 127-s − 135-s − 136-s + 155-s − 157-s + 163-s + 2·169-s − 184-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s)=\mathstrut & 26569 ^{s/2} \, \Gamma_{\R}(s)^{3} \, L(s,\rho)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(26569\) = \(163^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 26569,\ (0, 0, 0:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,\rho) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Particular Values
\[L(1/2,\rho) \approx 0.3724739457\]
\[L(1,\rho) \approx 0.7710042337\]