| L(s) = 1 | + 2-s + 3·3-s + 4-s + 5-s + 3·6-s + 3·7-s + 8-s + 6·9-s + 10-s + 11-s + 3·12-s − 6·13-s + 3·14-s + 3·15-s + 16-s − 2·17-s + 6·18-s − 8·19-s + 20-s + 9·21-s + 22-s + 8·23-s + 3·24-s + 25-s − 6·26-s + 9·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 1.13·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 0.301·11-s + 0.866·12-s − 1.66·13-s + 0.801·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s + 1.41·18-s − 1.83·19-s + 0.223·20-s + 1.96·21-s + 0.213·22-s + 1.66·23-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 1.73·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.19676195478821, −13.58790898342947, −13.05287034884728, −12.76056634277165, −12.35036319594618, −11.55640924923346, −11.07941164001386, −10.48894548190499, −10.11116370053389, −9.340865289053838, −8.979353760275522, −8.591700754568021, −8.020455650531728, −7.433248513061049, −7.055740945131127, −6.628154634731643, −5.750579244986391, −4.954413376174514, −4.671350341007062, −4.290532100477362, −3.324885871905414, −3.072403878250258, −2.248939507268436, −1.831438718402718, −1.571437694287410, 0,
1.571437694287410, 1.831438718402718, 2.248939507268436, 3.072403878250258, 3.324885871905414, 4.290532100477362, 4.671350341007062, 4.954413376174514, 5.750579244986391, 6.628154634731643, 7.055740945131127, 7.433248513061049, 8.020455650531728, 8.591700754568021, 8.979353760275522, 9.340865289053838, 10.11116370053389, 10.48894548190499, 11.07941164001386, 11.55640924923346, 12.35036319594618, 12.76056634277165, 13.05287034884728, 13.58790898342947, 14.19676195478821
