| L(s) = 1 | + 5.21·2-s − 11.2·3-s − 4.81·4-s − 9.80·5-s − 58.7·6-s − 11.1·7-s − 191.·8-s − 116.·9-s − 51.1·10-s − 557.·11-s + 54.2·12-s + 107.·13-s − 57.8·14-s + 110.·15-s − 846.·16-s + 329.·17-s − 604.·18-s + 2.93e3·19-s + 47.2·20-s + 125.·21-s − 2.90e3·22-s − 385.·23-s + 2.16e3·24-s − 3.02e3·25-s + 561.·26-s + 4.04e3·27-s + 53.4·28-s + ⋯ |
| L(s) = 1 | + 0.921·2-s − 0.722·3-s − 0.150·4-s − 0.175·5-s − 0.666·6-s − 0.0856·7-s − 1.06·8-s − 0.477·9-s − 0.161·10-s − 1.38·11-s + 0.108·12-s + 0.176·13-s − 0.0789·14-s + 0.126·15-s − 0.826·16-s + 0.276·17-s − 0.440·18-s + 1.86·19-s + 0.0263·20-s + 0.0619·21-s − 1.27·22-s − 0.151·23-s + 0.766·24-s − 0.969·25-s + 0.162·26-s + 1.06·27-s + 0.0128·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 - 5.21T + 32T^{2} \) |
| 3 | \( 1 + 11.2T + 243T^{2} \) |
| 5 | \( 1 + 9.80T + 3.12e3T^{2} \) |
| 7 | \( 1 + 11.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 557.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 107.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 329.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.93e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 385.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.06e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 8.99e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.51e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.31e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.85e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 231.T + 8.58e9T^{2} \) |
| 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
−14.16994489482100625151263405823, −13.20729832047387833461415633161, −12.11366619875816739699463872328, −11.11158401511466941167917805765, −9.539109692263321289514837004048, −7.82015069556346911211468700734, −5.87204909189281793535246455928, −5.06101921749643034890107866414, −3.22372600752869639428883370053, 0,
3.22372600752869639428883370053, 5.06101921749643034890107866414, 5.87204909189281793535246455928, 7.82015069556346911211468700734, 9.539109692263321289514837004048, 11.11158401511466941167917805765, 12.11366619875816739699463872328, 13.20729832047387833461415633161, 14.16994489482100625151263405823
