L(s) = 1 | − 2.73·2-s + 3-s + 5.47·4-s − 3.64·5-s − 2.73·6-s − 3.96·7-s − 9.49·8-s + 9-s + 9.96·10-s + 11-s + 5.47·12-s + 2.49·13-s + 10.8·14-s − 3.64·15-s + 14.9·16-s − 4.84·17-s − 2.73·18-s − 0.0465·19-s − 19.9·20-s − 3.96·21-s − 2.73·22-s + 7.87·23-s − 9.49·24-s + 8.30·25-s − 6.81·26-s + 27-s − 21.6·28-s + ⋯ |
L(s) = 1 | − 1.93·2-s + 0.577·3-s + 2.73·4-s − 1.63·5-s − 1.11·6-s − 1.49·7-s − 3.35·8-s + 0.333·9-s + 3.15·10-s + 0.301·11-s + 1.57·12-s + 0.691·13-s + 2.89·14-s − 0.941·15-s + 3.74·16-s − 1.17·17-s − 0.644·18-s − 0.0106·19-s − 4.46·20-s − 0.864·21-s − 0.582·22-s + 1.64·23-s − 1.93·24-s + 1.66·25-s − 1.33·26-s + 0.192·27-s − 4.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 17 | \( 1 + 4.84T + 17T^{2} \) |
| 19 | \( 1 + 0.0465T + 19T^{2} \) |
| 23 | \( 1 - 7.87T + 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 - 6.56T + 43T^{2} \) |
| 47 | \( 1 - 0.000570T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 + 7.63T + 59T^{2} \) |
| 67 | \( 1 + 8.92T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 8.55T + 79T^{2} \) |
| 83 | \( 1 - 3.71T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 6.11T + 97T^{2} \) |
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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
−8.750520696770591140398285401526, −8.263654375972811654656358146105, −7.37222564282430485572232515333, −6.78599327827427905738209226489, −6.30648875761418726939987016509, −4.36155864319286443386223699656, −3.21577028503659093326318284442, −2.82149408645607506802514439416, −1.10574246248730747778597475428, 0,
1.10574246248730747778597475428, 2.82149408645607506802514439416, 3.21577028503659093326318284442, 4.36155864319286443386223699656, 6.30648875761418726939987016509, 6.78599327827427905738209226489, 7.37222564282430485572232515333, 8.263654375972811654656358146105, 8.750520696770591140398285401526