L(s) = 1 | − 2-s − 0.715·3-s + 4-s + 2.28·5-s + 0.715·6-s − 4.66·7-s − 8-s − 2.48·9-s − 2.28·10-s + 3.56·11-s − 0.715·12-s + 2.24·13-s + 4.66·14-s − 1.63·15-s + 16-s + 0.341·17-s + 2.48·18-s − 2.98·19-s + 2.28·20-s + 3.34·21-s − 3.56·22-s + 0.277·23-s + 0.715·24-s + 0.205·25-s − 2.24·26-s + 3.92·27-s − 4.66·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.413·3-s + 0.5·4-s + 1.02·5-s + 0.292·6-s − 1.76·7-s − 0.353·8-s − 0.829·9-s − 0.721·10-s + 1.07·11-s − 0.206·12-s + 0.621·13-s + 1.24·14-s − 0.421·15-s + 0.250·16-s + 0.0828·17-s + 0.586·18-s − 0.685·19-s + 0.510·20-s + 0.729·21-s − 0.761·22-s + 0.0578·23-s + 0.146·24-s + 0.0410·25-s − 0.439·26-s + 0.755·27-s − 0.882·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 0.715T + 3T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 0.341T + 17T^{2} \) |
| 19 | \( 1 + 2.98T + 19T^{2} \) |
| 23 | \( 1 - 0.277T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 - 0.730T + 41T^{2} \) |
| 43 | \( 1 - 4.51T + 43T^{2} \) |
| 47 | \( 1 + 6.11T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 + 3.99T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 1.09T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 6.25T + 83T^{2} \) |
| 89 | \( 1 + 5.40T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−8.373748723531008248032908744769, −7.09589154548590329075265698660, −6.44947181386862481349374664455, −6.10598172230138256643558047881, −5.53821753250745708330668725638, −4.10557733598863442688026321262, −3.20448819347477277827673218558, −2.42810645687449579687241836529, −1.22683196190181123639492662763, 0,
1.22683196190181123639492662763, 2.42810645687449579687241836529, 3.20448819347477277827673218558, 4.10557733598863442688026321262, 5.53821753250745708330668725638, 6.10598172230138256643558047881, 6.44947181386862481349374664455, 7.09589154548590329075265698660, 8.373748723531008248032908744769