Dirichlet series
| $L(s,\rho)$ = 1 | − 2-s − 7-s − 13-s + 14-s + 16-s + 23-s + 26-s + 27-s + 29-s − 31-s − 32-s − 37-s − 41-s − 46-s + 47-s − 53-s − 54-s − 58-s − 59-s + 62-s − 67-s + 73-s + 74-s + 79-s + 82-s + 89-s + 91-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(229\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 229,\ (0, 1, 1:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,\rho) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Particular Values
\[L(1/2,\rho) \approx 0.2157489918\]
\[L(1,\rho) \approx 0.4400768328\]