| L(s) = 1 | − 2-s + 0.191·3-s + 4-s − 1.50·5-s − 0.191·6-s − 2.08·7-s − 8-s − 2.96·9-s + 1.50·10-s + 5.27·11-s + 0.191·12-s − 2.65·13-s + 2.08·14-s − 0.289·15-s + 16-s + 1.37·17-s + 2.96·18-s + 4.07·19-s − 1.50·20-s − 0.399·21-s − 5.27·22-s + 0.484·23-s − 0.191·24-s − 2.72·25-s + 2.65·26-s − 1.14·27-s − 2.08·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.110·3-s + 0.5·4-s − 0.673·5-s − 0.0783·6-s − 0.786·7-s − 0.353·8-s − 0.987·9-s + 0.476·10-s + 1.59·11-s + 0.0554·12-s − 0.735·13-s + 0.556·14-s − 0.0746·15-s + 0.250·16-s + 0.334·17-s + 0.698·18-s + 0.935·19-s − 0.336·20-s − 0.0872·21-s − 1.12·22-s + 0.101·23-s − 0.0391·24-s − 0.545·25-s + 0.520·26-s − 0.220·27-s − 0.393·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 0.191T + 3T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 4.07T + 19T^{2} \) |
| 23 | \( 1 - 0.484T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 37 | \( 1 - 8.02T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 - 0.0207T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 + 8.31T + 67T^{2} \) |
| 71 | \( 1 + 6.45T + 71T^{2} \) |
| 73 | \( 1 - 2.97T + 73T^{2} \) |
| 79 | \( 1 - 6.72T + 79T^{2} \) |
| 83 | \( 1 + 9.10T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
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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
−7.61832955035860290207459337130, −7.28451701348701733688754266304, −6.32448612498894121459995080779, −5.89007473343530372369464809691, −4.81120189734624312850020370092, −3.73892802498136010505510643604, −3.27822194518427470697618189137, −2.32502374896101808128607091345, −1.08233954903686033696577716341, 0,
1.08233954903686033696577716341, 2.32502374896101808128607091345, 3.27822194518427470697618189137, 3.73892802498136010505510643604, 4.81120189734624312850020370092, 5.89007473343530372369464809691, 6.32448612498894121459995080779, 7.28451701348701733688754266304, 7.61832955035860290207459337130
