| L(s) = 1 | − 2-s + 2.23·3-s + 4-s + 0.390·5-s − 2.23·6-s − 4.11·7-s − 8-s + 2.00·9-s − 0.390·10-s + 0.0299·11-s + 2.23·12-s − 2.40·13-s + 4.11·14-s + 0.873·15-s + 16-s + 1.64·17-s − 2.00·18-s + 6.28·19-s + 0.390·20-s − 9.19·21-s − 0.0299·22-s + 1.64·23-s − 2.23·24-s − 4.84·25-s + 2.40·26-s − 2.22·27-s − 4.11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.174·5-s − 0.913·6-s − 1.55·7-s − 0.353·8-s + 0.667·9-s − 0.123·10-s + 0.00902·11-s + 0.645·12-s − 0.666·13-s + 1.09·14-s + 0.225·15-s + 0.250·16-s + 0.398·17-s − 0.472·18-s + 1.44·19-s + 0.0873·20-s − 2.00·21-s − 0.00637·22-s + 0.344·23-s − 0.456·24-s − 0.969·25-s + 0.471·26-s − 0.429·27-s − 0.776·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 0.390T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 - 0.0299T + 11T^{2} \) |
| 13 | \( 1 + 2.40T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 + 4.33T + 31T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + 9.74T + 53T^{2} \) |
| 59 | \( 1 + 0.526T + 59T^{2} \) |
| 61 | \( 1 + 2.37T + 61T^{2} \) |
| 67 | \( 1 - 7.16T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 5.86T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 - 8.38T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.033698389406680794518973587399, −7.57147134514996119994862095447, −6.84659607355742812124660618549, −6.07801928037472176905074603777, −5.17612572960807078820157888686, −3.80741236202729360681829909913, −3.12112431913004510910119369771, −2.66918540679573813751208425104, −1.51283327771816596590866903348, 0,
1.51283327771816596590866903348, 2.66918540679573813751208425104, 3.12112431913004510910119369771, 3.80741236202729360681829909913, 5.17612572960807078820157888686, 6.07801928037472176905074603777, 6.84659607355742812124660618549, 7.57147134514996119994862095447, 8.033698389406680794518973587399
