| L(s) = 1 | − 2·3-s + 1.37·5-s + 3.37·7-s + 9-s + 2·11-s − 6.74·13-s − 2.74·15-s − 2·17-s + 0.627·19-s − 6.74·21-s − 6.74·23-s − 3.11·25-s + 4·27-s − 8·29-s − 31-s − 4·33-s + 4.62·35-s + 6.74·37-s + 13.4·39-s + 1.37·41-s + 4.74·43-s + 1.37·45-s + 4.37·49-s + 4·51-s + 2.74·55-s − 1.25·57-s + 3.37·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.613·5-s + 1.27·7-s + 0.333·9-s + 0.603·11-s − 1.87·13-s − 0.708·15-s − 0.485·17-s + 0.144·19-s − 1.47·21-s − 1.40·23-s − 0.623·25-s + 0.769·27-s − 1.48·29-s − 0.179·31-s − 0.696·33-s + 0.782·35-s + 1.10·37-s + 2.15·39-s + 0.214·41-s + 0.723·43-s + 0.204·45-s + 0.624·49-s + 0.560·51-s + 0.370·55-s − 0.166·57-s + 0.439·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 37 | \( 1 - 6.74T + 37T^{2} \) |
| 41 | \( 1 - 1.37T + 41T^{2} \) |
| 43 | \( 1 - 4.74T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 3.37T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 8.74T + 89T^{2} \) |
| 97 | \( 1 - 0.116T + 97T^{2} \) |
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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
−8.889677826374363436663515944909, −7.76748446426564412808500070591, −7.26889589032955982168711849273, −6.10833188179500883462055166062, −5.64895077327935162333522057245, −4.81044264447537159269022685761, −4.19473140252903069283378206725, −2.45319443324797614447375702784, −1.60428491081981852492702894437, 0,
1.60428491081981852492702894437, 2.45319443324797614447375702784, 4.19473140252903069283378206725, 4.81044264447537159269022685761, 5.64895077327935162333522057245, 6.10833188179500883462055166062, 7.26889589032955982168711849273, 7.76748446426564412808500070591, 8.889677826374363436663515944909
