L(s) = 1 | + 2-s − 2.41·3-s + 4-s − 1.28·5-s − 2.41·6-s + 2.70·7-s + 8-s + 2.82·9-s − 1.28·10-s + 0.288·11-s − 2.41·12-s − 6.11·13-s + 2.70·14-s + 3.11·15-s + 16-s − 17-s + 2.82·18-s + 3.39·19-s − 1.28·20-s − 6.52·21-s + 0.288·22-s − 1.12·23-s − 2.41·24-s − 3.34·25-s − 6.11·26-s + 0.414·27-s + 2.70·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.576·5-s − 0.985·6-s + 1.02·7-s + 0.353·8-s + 0.942·9-s − 0.407·10-s + 0.0869·11-s − 0.696·12-s − 1.69·13-s + 0.722·14-s + 0.803·15-s + 0.250·16-s − 0.242·17-s + 0.666·18-s + 0.779·19-s − 0.288·20-s − 1.42·21-s + 0.0614·22-s − 0.234·23-s − 0.492·24-s − 0.668·25-s − 1.19·26-s + 0.0797·27-s + 0.510·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 - 0.288T + 11T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 9.24T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 7.45T + 43T^{2} \) |
| 47 | \( 1 + 4.90T + 47T^{2} \) |
| 53 | \( 1 - 1.18T + 53T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 + 9.22T + 67T^{2} \) |
| 71 | \( 1 + 9.08T + 71T^{2} \) |
| 73 | \( 1 - 0.591T + 73T^{2} \) |
| 79 | \( 1 + 7.88T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.623397127907027608640681233809, −7.59950885163688352979015537107, −7.18366460387994949435150032264, −6.20909778573985511408378905095, −5.32494569211008343836429373496, −4.85190875495597084280990807918, −4.20665843331795519758664882498, −2.85708932535055773630289143105, −1.54949206308104747706038746412, 0,
1.54949206308104747706038746412, 2.85708932535055773630289143105, 4.20665843331795519758664882498, 4.85190875495597084280990807918, 5.32494569211008343836429373496, 6.20909778573985511408378905095, 7.18366460387994949435150032264, 7.59950885163688352979015537107, 8.623397127907027608640681233809