| L(s) = 1 | + 1.19e10·2-s + 1.85e15·3-s + 1.06e20·4-s + 6.35e22·5-s + 2.22e25·6-s − 1.28e27·7-s + 8.36e29·8-s + 3.43e30·9-s + 7.61e32·10-s + 6.32e33·11-s + 1.97e35·12-s − 9.80e35·13-s − 1.54e37·14-s + 1.17e38·15-s + 6.08e39·16-s − 1.86e40·17-s + 4.11e40·18-s + 1.91e41·19-s + 6.78e42·20-s − 2.38e42·21-s + 7.58e43·22-s + 9.32e43·23-s + 1.54e45·24-s + 1.32e45·25-s − 1.17e46·26-s + 6.36e45·27-s − 1.37e47·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 0.577·3-s + 2.89·4-s + 1.22·5-s + 1.13·6-s − 0.440·7-s + 3.73·8-s + 0.333·9-s + 2.40·10-s + 0.903·11-s + 1.66·12-s − 0.614·13-s − 0.868·14-s + 0.704·15-s + 4.47·16-s − 1.90·17-s + 0.657·18-s + 0.529·19-s + 3.52·20-s − 0.254·21-s + 1.78·22-s + 0.516·23-s + 2.15·24-s + 0.489·25-s − 1.21·26-s + 0.192·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+65/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(33)\) |
\(\approx\) |
\(12.64496178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.64496178\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.85e15T \) |
| good | 2 | \( 1 - 1.19e10T + 3.68e19T^{2} \) |
| 5 | \( 1 - 6.35e22T + 2.71e45T^{2} \) |
| 7 | \( 1 + 1.28e27T + 8.53e54T^{2} \) |
| 11 | \( 1 - 6.32e33T + 4.90e67T^{2} \) |
| 13 | \( 1 + 9.80e35T + 2.54e72T^{2} \) |
| 17 | \( 1 + 1.86e40T + 9.53e79T^{2} \) |
| 19 | \( 1 - 1.91e41T + 1.31e83T^{2} \) |
| 23 | \( 1 - 9.32e43T + 3.25e88T^{2} \) |
| 29 | \( 1 + 4.35e47T + 1.13e95T^{2} \) |
| 31 | \( 1 + 8.83e47T + 8.67e96T^{2} \) |
| 37 | \( 1 + 7.54e50T + 8.57e101T^{2} \) |
| 41 | \( 1 - 8.49e51T + 6.77e104T^{2} \) |
| 43 | \( 1 - 3.99e52T + 1.49e106T^{2} \) |
| 47 | \( 1 - 1.24e54T + 4.85e108T^{2} \) |
| 53 | \( 1 + 1.31e56T + 1.19e112T^{2} \) |
| 59 | \( 1 + 3.08e57T + 1.27e115T^{2} \) |
| 61 | \( 1 - 2.37e57T + 1.11e116T^{2} \) |
| 67 | \( 1 - 2.28e59T + 4.95e118T^{2} \) |
| 71 | \( 1 - 2.50e60T + 2.14e120T^{2} \) |
| 73 | \( 1 + 1.74e60T + 1.30e121T^{2} \) |
| 79 | \( 1 + 2.56e60T + 2.21e123T^{2} \) |
| 83 | \( 1 - 2.86e62T + 5.49e124T^{2} \) |
| 89 | \( 1 + 2.78e63T + 5.13e126T^{2} \) |
| 97 | \( 1 + 3.87e63T + 1.38e129T^{2} \) |
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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
−13.54788123832408923443327311653, −12.66932311110569067933411673337, −11.08966532353986848143573388262, −9.468515680833281877796558800572, −7.06624473518344253851073856987, −6.17390857725875718106663095077, −4.90969088114051496638591555968, −3.67133610237072394391984520135, −2.45458821858320283230936286468, −1.69367592205243487771671221798,
1.69367592205243487771671221798, 2.45458821858320283230936286468, 3.67133610237072394391984520135, 4.90969088114051496638591555968, 6.17390857725875718106663095077, 7.06624473518344253851073856987, 9.468515680833281877796558800572, 11.08966532353986848143573388262, 12.66932311110569067933411673337, 13.54788123832408923443327311653
