Dirichlet series
| $L(s,\rho)$ = 1 | − 5-s − 7-s − 16-s + 23-s + 29-s − 31-s + 35-s + 2·47-s + 49-s − 53-s + 80-s − 81-s − 83-s − 89-s + 97-s − 101-s + 2·107-s + 112-s − 115-s − 2·127-s + 131-s − 139-s − 145-s − 149-s + 155-s − 161-s + 193-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s)=\mathstrut & 3215 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(3215\) = \(5 \cdot 643\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 3215,\ (0, 1, 1, 1:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,\rho) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Particular Values
\[L(1/2,\rho) \approx 0.3786092567\]
\[L(1,\rho) \approx 0.6887413040\]