Dirichlet series
| $L(s,\rho)$ = 1 | + 3-s + 8-s + 9-s − 17-s + 19-s + 24-s + 27-s − 37-s − 51-s − 53-s + 57-s + 64-s − 71-s + 72-s − 73-s + 81-s − 89-s − 107-s + 3·109-s − 111-s + 125-s − 127-s − 136-s + 152-s − 153-s − 159-s − 163-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(29241\) = \(3^{4} \cdot 19^{2}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((3,\ 29241,\ (0, 1, 1:\ ),\ 1)\) |
Particular Values
\[L(1/2,\rho) \approx 2.008121103\]
\[L(1,\rho) \approx 1.574301910\]
Euler product
\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line