| L(s) = 1 | − 1.08e3·2-s − 9.14e5·4-s − 1.49e6·5-s − 7.63e7·7-s + 3.27e9·8-s + 1.62e9·10-s + 1.45e11·11-s − 6.28e11·13-s + 8.30e10·14-s − 1.64e12·16-s + 8.35e12·17-s − 8.83e12·19-s + 1.36e12·20-s − 1.58e14·22-s + 8.45e13·23-s − 4.74e14·25-s + 6.83e14·26-s + 6.98e13·28-s − 1.39e15·29-s − 7.21e15·31-s − 5.08e15·32-s − 9.08e15·34-s + 1.14e14·35-s + 1.51e16·37-s + 9.60e15·38-s − 4.90e15·40-s + 7.57e16·41-s + ⋯ |
| L(s) = 1 | − 0.750·2-s − 0.436·4-s − 0.0685·5-s − 0.102·7-s + 1.07·8-s + 0.0514·10-s + 1.69·11-s − 1.26·13-s + 0.0767·14-s − 0.373·16-s + 1.00·17-s − 0.330·19-s + 0.0299·20-s − 1.27·22-s + 0.425·23-s − 0.995·25-s + 0.949·26-s + 0.0445·28-s − 0.617·29-s − 1.58·31-s − 0.798·32-s − 0.754·34-s + 0.00700·35-s + 0.517·37-s + 0.248·38-s − 0.0739·40-s + 0.880·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.08e3T + 2.09e6T^{2} \) |
| 5 | \( 1 + 1.49e6T + 4.76e14T^{2} \) |
| 7 | \( 1 + 7.63e7T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.45e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 6.28e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 8.35e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 8.83e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 8.45e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.39e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 7.21e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.51e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.57e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.45e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.38e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.10e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.24e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.77e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 3.50e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.49e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 6.36e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 2.99e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.77e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.84e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 9.03e20T + 5.27e41T^{2} \) |
| 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
−9.503215573795402419072287664212, −9.317276852710097516844000641969, −7.908162087720934051666265232171, −7.11308506880470460076981180648, −5.72725251496519467044299776068, −4.47270130899632928891991507928, −3.58349377108754024792755624135, −1.97243639745561411072808718274, −1.00531330890919917642355584497, 0,
1.00531330890919917642355584497, 1.97243639745561411072808718274, 3.58349377108754024792755624135, 4.47270130899632928891991507928, 5.72725251496519467044299776068, 7.11308506880470460076981180648, 7.908162087720934051666265232171, 9.317276852710097516844000641969, 9.503215573795402419072287664212
