| L(s) = 1 | − 2-s − 3.25·3-s + 4-s − 3.41·5-s + 3.25·6-s + 1.95·7-s − 8-s + 7.58·9-s + 3.41·10-s + 2.17·11-s − 3.25·12-s − 3.20·13-s − 1.95·14-s + 11.1·15-s + 16-s − 4.06·17-s − 7.58·18-s − 4.62·19-s − 3.41·20-s − 6.34·21-s − 2.17·22-s + 3.12·23-s + 3.25·24-s + 6.69·25-s + 3.20·26-s − 14.8·27-s + 1.95·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.87·3-s + 0.5·4-s − 1.52·5-s + 1.32·6-s + 0.737·7-s − 0.353·8-s + 2.52·9-s + 1.08·10-s + 0.656·11-s − 0.938·12-s − 0.887·13-s − 0.521·14-s + 2.87·15-s + 0.250·16-s − 0.985·17-s − 1.78·18-s − 1.06·19-s − 0.764·20-s − 1.38·21-s − 0.464·22-s + 0.652·23-s + 0.663·24-s + 1.33·25-s + 0.627·26-s − 2.86·27-s + 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 + 3.20T + 13T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 + 4.62T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + 5.84T + 31T^{2} \) |
| 37 | \( 1 - 3.29T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 + 6.37T + 53T^{2} \) |
| 59 | \( 1 + 0.661T + 59T^{2} \) |
| 61 | \( 1 - 2.59T + 61T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 4.40T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 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
−7.87980335262339290195668578802, −7.31044779372886760478235975323, −6.70393687530015311264397777186, −6.07510793018864194374643140123, −4.80570530276194857030894739759, −4.64241341149845196659761102697, −3.69733602321912357369448628779, −2.06452598668172252235786722656, −0.867564473633110727107348514902, 0,
0.867564473633110727107348514902, 2.06452598668172252235786722656, 3.69733602321912357369448628779, 4.64241341149845196659761102697, 4.80570530276194857030894739759, 6.07510793018864194374643140123, 6.70393687530015311264397777186, 7.31044779372886760478235975323, 7.87980335262339290195668578802
