| L(s) = 1 | + 6.22e4·3-s + 1.24e6·5-s + 2.63e8·7-s − 6.58e9·9-s − 3.37e10·11-s − 2.50e11·13-s + 7.75e10·15-s + 2.34e11·17-s − 3.24e13·19-s + 1.63e13·21-s + 9.63e13·23-s − 4.75e14·25-s − 1.06e15·27-s + 3.86e15·29-s + 2.26e15·31-s − 2.09e15·33-s + 3.27e14·35-s − 4.75e15·37-s − 1.56e16·39-s + 3.56e16·41-s + 2.20e17·43-s − 8.20e15·45-s − 4.79e17·47-s − 4.89e17·49-s + 1.46e16·51-s + 1.86e18·53-s − 4.19e16·55-s + ⋯ |
| L(s) = 1 | + 0.608·3-s + 0.0570·5-s + 0.351·7-s − 0.629·9-s − 0.391·11-s − 0.504·13-s + 0.0347·15-s + 0.0282·17-s − 1.21·19-s + 0.214·21-s + 0.484·23-s − 0.996·25-s − 0.991·27-s + 1.70·29-s + 0.497·31-s − 0.238·33-s + 0.0200·35-s − 0.162·37-s − 0.307·39-s + 0.414·41-s + 1.55·43-s − 0.0359·45-s − 1.33·47-s − 0.876·49-s + 0.0171·51-s + 1.46·53-s − 0.0223·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.251663734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.251663734\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 6.22e4T + 1.04e10T^{2} \) |
| 5 | \( 1 - 1.24e6T + 4.76e14T^{2} \) |
| 7 | \( 1 - 2.63e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.37e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 2.50e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.34e11T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.24e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 9.63e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.86e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 2.26e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 4.75e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 3.56e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.20e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.79e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.86e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 1.23e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 6.27e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.72e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.13e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 5.49e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 8.34e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.22e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.21e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 3.73e20T + 5.27e41T^{2} \) |
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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
−10.88597695593321023699550620730, −9.769773121567286165520878550247, −8.589845587635188152189073409972, −7.905381000936628771743230317440, −6.54636941188842671765589761886, −5.31080574529752454719461024020, −4.15389851138012206902918767382, −2.83318931769387687807282483514, −2.08051902969554027680736771574, −0.60015290648601372587670666729,
0.60015290648601372587670666729, 2.08051902969554027680736771574, 2.83318931769387687807282483514, 4.15389851138012206902918767382, 5.31080574529752454719461024020, 6.54636941188842671765589761886, 7.905381000936628771743230317440, 8.589845587635188152189073409972, 9.769773121567286165520878550247, 10.88597695593321023699550620730
