| L(s) = 1 | + 1.87·2-s + 1.52·4-s − 3.86·5-s − 2.13·7-s − 0.899·8-s − 7.25·10-s + 3.11·11-s + 2.25·13-s − 4.01·14-s − 4.72·16-s + 2.97·17-s + 19-s − 5.88·20-s + 5.84·22-s + 4.45·23-s + 9.95·25-s + 4.23·26-s − 3.25·28-s + 0.450·29-s + 4.13·31-s − 7.07·32-s + 5.57·34-s + 8.27·35-s − 2.09·37-s + 1.87·38-s + 3.47·40-s − 7.08·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.760·4-s − 1.72·5-s − 0.808·7-s − 0.317·8-s − 2.29·10-s + 0.939·11-s + 0.625·13-s − 1.07·14-s − 1.18·16-s + 0.720·17-s + 0.229·19-s − 1.31·20-s + 1.24·22-s + 0.928·23-s + 1.99·25-s + 0.829·26-s − 0.614·28-s + 0.0835·29-s + 0.742·31-s − 1.25·32-s + 0.955·34-s + 1.39·35-s − 0.343·37-s + 0.304·38-s + 0.549·40-s − 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 5 | \( 1 + 3.86T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 - 2.97T + 17T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 - 0.450T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 41 | \( 1 + 7.08T + 41T^{2} \) |
| 43 | \( 1 + 6.87T + 43T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 + 0.860T + 59T^{2} \) |
| 61 | \( 1 + 3.35T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 3.83T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.18983353244078395703398897189, −6.71559844056254958692163787989, −6.11899228459377482178961490236, −5.15803533690910768230309520029, −4.55281800419192364699371964643, −3.76471907022780208040489100928, −3.44566759316990625691328650267, −2.87743745032671473412955691857, −1.21412320794033808182465078430, 0,
1.21412320794033808182465078430, 2.87743745032671473412955691857, 3.44566759316990625691328650267, 3.76471907022780208040489100928, 4.55281800419192364699371964643, 5.15803533690910768230309520029, 6.11899228459377482178961490236, 6.71559844056254958692163787989, 7.18983353244078395703398897189
