| L(s) = 1 | + 2-s + 1.65·3-s + 4-s + 1.27·5-s + 1.65·6-s − 2.65·7-s + 8-s − 0.273·9-s + 1.27·10-s − 3.27·11-s + 1.65·12-s − 2.65·14-s + 2.10·15-s + 16-s − 5.30·17-s − 0.273·18-s + 19-s + 1.27·20-s − 4.37·21-s − 3.27·22-s + 6.40·23-s + 1.65·24-s − 3.37·25-s − 5.40·27-s − 2.65·28-s − 7.48·29-s + 2.10·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.953·3-s + 0.5·4-s + 0.569·5-s + 0.674·6-s − 1.00·7-s + 0.353·8-s − 0.0912·9-s + 0.402·10-s − 0.987·11-s + 0.476·12-s − 0.708·14-s + 0.543·15-s + 0.250·16-s − 1.28·17-s − 0.0645·18-s + 0.229·19-s + 0.284·20-s − 0.955·21-s − 0.697·22-s + 1.33·23-s + 0.337·24-s − 0.675·25-s − 1.04·27-s − 0.501·28-s − 1.38·29-s + 0.384·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.65T + 3T^{2} \) |
| 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 + 2.65T + 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 23 | \( 1 - 6.40T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 2.20T + 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 2.47T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 + 5.67T + 61T^{2} \) |
| 67 | \( 1 + 0.206T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 7.79T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.62560960166538454926323736714, −6.88852847107395876350750786855, −6.22898555292719715285877859988, −5.52099739220083971780658402990, −4.81341827676471401596396602021, −3.81491596962926872572071942873, −3.08759137113016327027651779366, −2.58176134129582938742176235720, −1.79547518476637143584703761611, 0,
1.79547518476637143584703761611, 2.58176134129582938742176235720, 3.08759137113016327027651779366, 3.81491596962926872572071942873, 4.81341827676471401596396602021, 5.52099739220083971780658402990, 6.22898555292719715285877859988, 6.88852847107395876350750786855, 7.62560960166538454926323736714
