| L(s) = 1 | − 2-s + 1.40·3-s + 4-s − 5-s − 1.40·6-s − 2.31·7-s − 8-s − 1.01·9-s + 10-s + 3.47·11-s + 1.40·12-s − 0.582·13-s + 2.31·14-s − 1.40·15-s + 16-s − 0.139·17-s + 1.01·18-s + 1.62·19-s − 20-s − 3.25·21-s − 3.47·22-s + 3.15·23-s − 1.40·24-s + 25-s + 0.582·26-s − 5.65·27-s − 2.31·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.812·3-s + 0.5·4-s − 0.447·5-s − 0.574·6-s − 0.874·7-s − 0.353·8-s − 0.339·9-s + 0.316·10-s + 1.04·11-s + 0.406·12-s − 0.161·13-s + 0.618·14-s − 0.363·15-s + 0.250·16-s − 0.0337·17-s + 0.240·18-s + 0.373·19-s − 0.223·20-s − 0.710·21-s − 0.741·22-s + 0.657·23-s − 0.287·24-s + 0.200·25-s + 0.114·26-s − 1.08·27-s − 0.437·28-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 | \( 1 - 1.40T + 3T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 + 0.582T + 13T^{2} \) |
| 17 | \( 1 + 0.139T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 3.15T + 23T^{2} \) |
| 29 | \( 1 + 0.911T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 + 9.08T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 0.278T + 47T^{2} \) |
| 53 | \( 1 + 4.00T + 53T^{2} \) |
| 59 | \( 1 + 2.63T + 59T^{2} \) |
| 61 | \( 1 - 6.69T + 61T^{2} \) |
| 67 | \( 1 + 3.55T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 4.01T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 + 5.16T + 89T^{2} \) |
| 97 | \( 1 + 3.28T + 97T^{2} \) |
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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
−8.269952610786503317238695060700, −7.41597043749303324817789194384, −6.83259524827549041045397472083, −6.12765379287714230493633523043, −5.11904798069618043906183835507, −3.87505732211414054390657693021, −3.31710947618887203779940928831, −2.55137831998153904216098095816, −1.37034349656445976794769453856, 0,
1.37034349656445976794769453856, 2.55137831998153904216098095816, 3.31710947618887203779940928831, 3.87505732211414054390657693021, 5.11904798069618043906183835507, 6.12765379287714230493633523043, 6.83259524827549041045397472083, 7.41597043749303324817789194384, 8.269952610786503317238695060700
