Dirichlet series
| $L(s,f)$ = 1 | + 2.06·2-s + 1.79·3-s + 1.79·4-s + 0.454·5-s + 3.71·6-s + 0.0379·7-s + 0.675·8-s + 0.779·9-s + 0.939·10-s − 0.992·11-s + 3.23·12-s − 0.332·13-s + 0.0785·14-s + 0.816·15-s + 0.219·16-s − 0.298·17-s + 1.61·18-s + 0.400·19-s + 0.817·20-s + 0.0682·21-s − 2.05·22-s + 0.698·23-s + 1.21·24-s + 1.03·25-s − 0.687·26-s − 1.21·27-s + 0.0682·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+19.8i) \, \Gamma_{\R}(s+8.53i) \, \Gamma_{\R}(s-19.8i) \, \Gamma_{\R}(s-8.53i) \, L(s,f)\cr
=\mathstrut & \,\Lambda(1-s,f)
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(1\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 1,\ (19.81936708928i, 8.53107821814i, -19.81936708928i, -8.53107821814i:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]