| L(s) = 1 | − 2.51·2-s − 2.51·3-s + 4.30·4-s + 3.86·5-s + 6.31·6-s + 2.24·7-s − 5.80·8-s + 3.32·9-s − 9.72·10-s − 3.66·11-s − 10.8·12-s − 4.71·13-s − 5.62·14-s − 9.72·15-s + 5.95·16-s + 5.95·17-s − 8.34·18-s + 19-s + 16.6·20-s − 5.63·21-s + 9.21·22-s − 5.33·23-s + 14.5·24-s + 9.97·25-s + 11.8·26-s − 0.806·27-s + 9.65·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 1.45·3-s + 2.15·4-s + 1.73·5-s + 2.57·6-s + 0.847·7-s − 2.05·8-s + 1.10·9-s − 3.07·10-s − 1.10·11-s − 3.12·12-s − 1.30·13-s − 1.50·14-s − 2.51·15-s + 1.48·16-s + 1.44·17-s − 1.96·18-s + 0.229·19-s + 3.72·20-s − 1.22·21-s + 1.96·22-s − 1.11·23-s + 2.97·24-s + 1.99·25-s + 2.32·26-s − 0.155·27-s + 1.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 + 2.51T + 3T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 - 5.95T + 17T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 41 | \( 1 + 8.57T + 41T^{2} \) |
| 43 | \( 1 + 1.69T + 43T^{2} \) |
| 47 | \( 1 - 0.701T + 47T^{2} \) |
| 53 | \( 1 - 0.437T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 0.0150T + 61T^{2} \) |
| 67 | \( 1 - 1.84T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 8.08T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 8.34T + 89T^{2} \) |
| 97 | \( 1 - 0.165T + 97T^{2} \) |
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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
−8.014279609196472542071163403504, −7.48334177690853716700687393532, −6.72246993805204618548394281579, −5.81839935600310169752174288192, −5.48829569871242279442196976587, −4.80712400965946847547632706931, −2.74436949666393731372271614299, −1.95465307910926072176671868859, −1.20763724249943144623597739678, 0,
1.20763724249943144623597739678, 1.95465307910926072176671868859, 2.74436949666393731372271614299, 4.80712400965946847547632706931, 5.48829569871242279442196976587, 5.81839935600310169752174288192, 6.72246993805204618548394281579, 7.48334177690853716700687393532, 8.014279609196472542071163403504
