| L(s) = 1 | + 2-s + 0.660·3-s + 4-s − 5-s + 0.660·6-s + 0.752·7-s + 8-s − 2.56·9-s − 10-s − 1.80·11-s + 0.660·12-s − 2.15·13-s + 0.752·14-s − 0.660·15-s + 16-s − 6.44·17-s − 2.56·18-s + 6.40·19-s − 20-s + 0.496·21-s − 1.80·22-s + 0.238·23-s + 0.660·24-s + 25-s − 2.15·26-s − 3.67·27-s + 0.752·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.381·3-s + 0.5·4-s − 0.447·5-s + 0.269·6-s + 0.284·7-s + 0.353·8-s − 0.854·9-s − 0.316·10-s − 0.543·11-s + 0.190·12-s − 0.598·13-s + 0.201·14-s − 0.170·15-s + 0.250·16-s − 1.56·17-s − 0.604·18-s + 1.46·19-s − 0.223·20-s + 0.108·21-s − 0.384·22-s + 0.0497·23-s + 0.134·24-s + 0.200·25-s − 0.422·26-s − 0.706·27-s + 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.660T + 3T^{2} \) |
| 7 | \( 1 - 0.752T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 - 6.40T + 19T^{2} \) |
| 23 | \( 1 - 0.238T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 - 0.727T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 4.78T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 + 5.98T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 + 7.95T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 8.98T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 9.45T + 89T^{2} \) |
| 97 | \( 1 - 6.49T + 97T^{2} \) |
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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
−8.010895786659203465220156376332, −7.36262196462089036526394001497, −6.61470009497974193526868946637, −5.73940098017905189565688133645, −4.89470159019721300735279977118, −4.45455119443262383785413422359, −3.16365345447012397856640903380, −2.85591668879680225001431037712, −1.70341702849131624979417957290, 0,
1.70341702849131624979417957290, 2.85591668879680225001431037712, 3.16365345447012397856640903380, 4.45455119443262383785413422359, 4.89470159019721300735279977118, 5.73940098017905189565688133645, 6.61470009497974193526868946637, 7.36262196462089036526394001497, 8.010895786659203465220156376332
