| L(s) = 1 | − 6.41e4·3-s + 9.76e6·5-s − 3.07e8·7-s − 6.34e9·9-s + 7.96e8·11-s − 5.08e11·13-s − 6.26e11·15-s + 1.49e12·17-s + 3.31e13·19-s + 1.97e13·21-s + 8.10e13·23-s + 9.53e13·25-s + 1.07e15·27-s − 8.66e14·29-s − 2.71e15·31-s − 5.11e13·33-s − 3.00e15·35-s + 4.01e16·37-s + 3.26e16·39-s − 6.20e16·41-s + 3.57e14·43-s − 6.19e16·45-s + 1.76e17·47-s − 4.63e17·49-s − 9.59e16·51-s − 3.44e17·53-s + 7.77e15·55-s + ⋯ |
| L(s) = 1 | − 0.627·3-s + 0.447·5-s − 0.411·7-s − 0.606·9-s + 0.00926·11-s − 1.02·13-s − 0.280·15-s + 0.179·17-s + 1.23·19-s + 0.258·21-s + 0.408·23-s + 0.199·25-s + 1.00·27-s − 0.382·29-s − 0.595·31-s − 0.00581·33-s − 0.184·35-s + 1.37·37-s + 0.642·39-s − 0.722·41-s + 0.00251·43-s − 0.271·45-s + 0.489·47-s − 0.830·49-s − 0.112·51-s − 0.270·53-s + 0.00414·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - 9.76e6T \) |
| good | 3 | \( 1 + 6.41e4T + 1.04e10T^{2} \) |
| 7 | \( 1 + 3.07e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 7.96e8T + 7.40e21T^{2} \) |
| 13 | \( 1 + 5.08e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.49e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.31e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 8.10e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 8.66e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.71e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.01e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 6.20e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 3.57e14T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.76e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 3.44e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.25e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 2.67e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.29e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.72e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 5.55e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 9.35e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.40e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.62e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 6.77e20T + 5.27e41T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.956615058773934570447901657319, −9.179929999116565913271600331942, −7.76160131453373600058470395229, −6.66785803899009654757038014501, −5.63829713089824024066332911560, −4.90879135714302341867435360562, −3.34777422296759433783185728000, −2.36904243866647237636733666693, −0.991102607111035239262604429703, 0,
0.991102607111035239262604429703, 2.36904243866647237636733666693, 3.34777422296759433783185728000, 4.90879135714302341867435360562, 5.63829713089824024066332911560, 6.66785803899009654757038014501, 7.76160131453373600058470395229, 9.179929999116565913271600331942, 9.956615058773934570447901657319
