Dirichlet series
| $L(s,f)$ = 1 | + 1.34·2-s − 0.187·3-s + 0.464·4-s − 0.00162·5-s − 0.251·6-s + 0.228·7-s + 0.169·8-s − 0.463·9-s − 0.00218·10-s + 0.695·11-s − 0.0870·12-s − 0.882·13-s + 0.306·14-s + 0.000304·15-s + 0.408·16-s + 0.716·17-s − 0.622·18-s − 0.927·19-s − 0.000699·20-s − 0.0427·21-s + 0.934·22-s + 0.419·23-s − 0.0309·24-s + 0.227·25-s − 1.17·26-s + 0.100·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+12.4i) \, \Gamma_{\R}(s+4.72i) \, \Gamma_{\R}(s-12.4i) \, \Gamma_{\R}(s-4.72i) \, L(s,f)\cr
=\mathstrut & \,\Lambda(1-s,f)
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(1\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 1,\ (12.46875226152i, 4.72095103638i, -12.46875226152i, -4.72095103638i:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]