| L(s) = 1 | + 9.28e4·3-s − 9.76e6·5-s − 9.95e8·7-s − 1.83e9·9-s + 8.11e10·11-s + 3.92e11·13-s − 9.06e11·15-s − 5.41e12·17-s + 2.09e13·19-s − 9.24e13·21-s − 4.40e13·23-s + 9.53e13·25-s − 1.14e15·27-s + 1.63e15·29-s − 3.46e14·31-s + 7.53e15·33-s + 9.71e15·35-s + 3.77e16·37-s + 3.64e16·39-s + 3.37e16·41-s + 7.72e16·43-s + 1.79e16·45-s − 3.30e17·47-s + 4.31e17·49-s − 5.02e17·51-s + 1.01e18·53-s − 7.92e17·55-s + ⋯ |
| L(s) = 1 | + 0.907·3-s − 0.447·5-s − 1.33·7-s − 0.175·9-s + 0.943·11-s + 0.790·13-s − 0.406·15-s − 0.651·17-s + 0.783·19-s − 1.20·21-s − 0.221·23-s + 0.199·25-s − 1.06·27-s + 0.723·29-s − 0.0759·31-s + 0.856·33-s + 0.595·35-s + 1.29·37-s + 0.717·39-s + 0.392·41-s + 0.544·43-s + 0.0785·45-s − 0.916·47-s + 0.772·49-s − 0.591·51-s + 0.793·53-s − 0.421·55-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 | \( 1 - 9.28e4T + 1.04e10T^{2} \) |
| 7 | \( 1 + 9.95e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.11e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 3.92e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 5.41e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.09e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 4.40e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.63e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 3.46e14T + 2.08e31T^{2} \) |
| 37 | \( 1 - 3.77e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 3.37e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 7.72e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.30e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.01e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 1.45e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.02e19T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.20e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.19e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.08e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 2.08e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.55e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.59e19T + 8.65e40T^{2} \) |
| 97 | \( 1 + 5.45e19T + 5.27e41T^{2} \) |
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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
−9.648041893593141542157386948277, −9.023102844831899818387884075420, −8.035687599340121071326314339330, −6.81170163948774306396379586696, −5.94174356329658032587267461727, −4.18783409468696970297614831357, −3.38927565098459266813178159751, −2.61845682994694176176364519863, −1.16810598363012226569409661041, 0,
1.16810598363012226569409661041, 2.61845682994694176176364519863, 3.38927565098459266813178159751, 4.18783409468696970297614831357, 5.94174356329658032587267461727, 6.81170163948774306396379586696, 8.035687599340121071326314339330, 9.023102844831899818387884075420, 9.648041893593141542157386948277
