| L(s) = 1 | − 4.11·2-s − 4.01·3-s + 8.90·4-s − 12.3·5-s + 16.5·6-s + 29.6·7-s − 3.74·8-s − 10.8·9-s + 50.7·10-s + 11·11-s − 35.7·12-s − 121.·14-s + 49.5·15-s − 55.8·16-s + 100.·17-s + 44.7·18-s + 38.7·19-s − 109.·20-s − 119.·21-s − 45.2·22-s − 107.·23-s + 15.0·24-s + 27.3·25-s + 152.·27-s + 264.·28-s + 136.·29-s − 203.·30-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.772·3-s + 1.11·4-s − 1.10·5-s + 1.12·6-s + 1.60·7-s − 0.165·8-s − 0.402·9-s + 1.60·10-s + 0.301·11-s − 0.860·12-s − 2.32·14-s + 0.853·15-s − 0.873·16-s + 1.42·17-s + 0.585·18-s + 0.468·19-s − 1.22·20-s − 1.23·21-s − 0.438·22-s − 0.977·23-s + 0.127·24-s + 0.218·25-s + 1.08·27-s + 1.78·28-s + 0.871·29-s − 1.24·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.11T + 8T^{2} \) |
| 3 | \( 1 + 4.01T + 27T^{2} \) |
| 5 | \( 1 + 12.3T + 125T^{2} \) |
| 7 | \( 1 - 29.6T + 343T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 7.90T + 2.97e4T^{2} \) |
| 37 | \( 1 + 415.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 257.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 19.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 167.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 96.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 520.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 702.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 517.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 77.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 545.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 663.T + 9.12e5T^{2} \) |
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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
−8.442737887322108551041034990055, −7.75316915360273498059624049485, −7.45228865944302094061449500634, −6.25895286914221049105495695105, −5.21356987757964664064133032800, −4.54132436474419834024754460649, −3.35461674541504574288282244174, −1.79869797146486667747929479221, −0.964894538857032601493909296148, 0,
0.964894538857032601493909296148, 1.79869797146486667747929479221, 3.35461674541504574288282244174, 4.54132436474419834024754460649, 5.21356987757964664064133032800, 6.25895286914221049105495695105, 7.45228865944302094061449500634, 7.75316915360273498059624049485, 8.442737887322108551041034990055
