| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 4·11-s + 4·13-s − 14-s + 16-s − 4·19-s + 20-s − 4·22-s − 23-s + 25-s + 4·26-s − 28-s − 2·29-s + 4·31-s + 32-s − 35-s − 2·37-s − 4·38-s + 40-s − 12·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 0.208·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.169·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 1.87·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.17514646033064, −15.65916796131118, −15.41383420354171, −14.72685746018497, −13.86539199152741, −13.63304636539790, −13.16430820594593, −12.45352561525051, −12.19690128793123, −11.08528696272671, −10.86191976003438, −10.26883454525285, −9.628216732249000, −8.862983591093002, −8.205538983832673, −7.708576857544008, −6.723183385845658, −6.378662950327905, −5.669222590080019, −5.142271089488863, −4.365918041939163, −3.631497899091796, −2.922385046497759, −2.226024555569938, −1.359990211474173, 0,
1.359990211474173, 2.226024555569938, 2.922385046497759, 3.631497899091796, 4.365918041939163, 5.142271089488863, 5.669222590080019, 6.378662950327905, 6.723183385845658, 7.708576857544008, 8.205538983832673, 8.862983591093002, 9.628216732249000, 10.26883454525285, 10.86191976003438, 11.08528696272671, 12.19690128793123, 12.45352561525051, 13.16430820594593, 13.63304636539790, 13.86539199152741, 14.72685746018497, 15.41383420354171, 15.65916796131118, 16.17514646033064
