| L(s) = 1 | + 5.29·5-s − 7·7-s − 68.7·11-s − 30·13-s + 121.·17-s − 100·19-s + 89.9·23-s − 97·25-s − 232.·29-s − 180·31-s − 37.0·35-s − 118·37-s − 15.8·41-s − 412·43-s + 285.·47-s + 49·49-s + 285.·53-s − 364·55-s − 836.·59-s − 378·61-s − 158.·65-s + 244·67-s + 439.·71-s + 670·73-s + 481.·77-s + 216·79-s + 804.·83-s + ⋯ |
| L(s) = 1 | + 0.473·5-s − 0.377·7-s − 1.88·11-s − 0.640·13-s + 1.73·17-s − 1.20·19-s + 0.815·23-s − 0.776·25-s − 1.49·29-s − 1.04·31-s − 0.178·35-s − 0.524·37-s − 0.0604·41-s − 1.46·43-s + 0.886·47-s + 0.142·49-s + 0.740·53-s − 0.892·55-s − 1.84·59-s − 0.793·61-s − 0.302·65-s + 0.444·67-s + 0.734·71-s + 1.07·73-s + 0.712·77-s + 0.307·79-s + 1.06·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 5.29T + 125T^{2} \) |
| 11 | \( 1 + 68.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 100T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 180T + 2.97e4T^{2} \) |
| 37 | \( 1 + 118T + 5.06e4T^{2} \) |
| 41 | \( 1 + 15.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 412T + 7.95e4T^{2} \) |
| 47 | \( 1 - 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 285.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 836.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 378T + 2.26e5T^{2} \) |
| 67 | \( 1 - 244T + 3.00e5T^{2} \) |
| 71 | \( 1 - 439.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 670T + 3.89e5T^{2} \) |
| 79 | \( 1 - 216T + 4.93e5T^{2} \) |
| 83 | \( 1 - 804.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 968.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 574T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.88989307596000546751255190741, −10.19955650144834699473275217139, −9.361726045799422160386089849100, −8.043272721330281910955615862944, −7.26809056783907814041640704283, −5.80683716741659530494741342430, −5.09819312603552985387440378572, −3.37775716309048803096757718830, −2.11403526358201869475263406389, 0,
2.11403526358201869475263406389, 3.37775716309048803096757718830, 5.09819312603552985387440378572, 5.80683716741659530494741342430, 7.26809056783907814041640704283, 8.043272721330281910955615862944, 9.361726045799422160386089849100, 10.19955650144834699473275217139, 10.88989307596000546751255190741
