| L(s) = 1 | + 8.37·2-s + 9·3-s + 38.1·4-s + 42.0·5-s + 75.3·6-s − 124.·7-s + 51.3·8-s + 81·9-s + 352.·10-s − 286.·11-s + 343.·12-s − 996.·13-s − 1.04e3·14-s + 378.·15-s − 789.·16-s − 530.·17-s + 678.·18-s − 968.·19-s + 1.60e3·20-s − 1.12e3·21-s − 2.40e3·22-s − 63.5·23-s + 462.·24-s − 1.35e3·25-s − 8.34e3·26-s + 729·27-s − 4.74e3·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 0.577·3-s + 1.19·4-s + 0.752·5-s + 0.854·6-s − 0.960·7-s + 0.283·8-s + 0.333·9-s + 1.11·10-s − 0.714·11-s + 0.688·12-s − 1.63·13-s − 1.42·14-s + 0.434·15-s − 0.771·16-s − 0.445·17-s + 0.493·18-s − 0.615·19-s + 0.896·20-s − 0.554·21-s − 1.05·22-s − 0.0250·23-s + 0.163·24-s − 0.433·25-s − 2.42·26-s + 0.192·27-s − 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 157 | \( 1 - 2.46e4T \) |
| good | 2 | \( 1 - 8.37T + 32T^{2} \) |
| 5 | \( 1 - 42.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 124.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 286.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 996.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 530.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 968.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 63.5T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.16e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.50e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.56e4T + 8.58e9T^{2} \) |
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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
−9.772098267560115135805584041789, −9.101312144111718010890406861137, −7.65914079233318588036272209279, −6.70235503347361284081915296642, −5.84946461515908432080842230197, −4.92565051296659281825147024604, −3.94925637460384894764177003762, −2.71834238070038930663585846920, −2.27603638125959835106531203179, 0,
2.27603638125959835106531203179, 2.71834238070038930663585846920, 3.94925637460384894764177003762, 4.92565051296659281825147024604, 5.84946461515908432080842230197, 6.70235503347361284081915296642, 7.65914079233318588036272209279, 9.101312144111718010890406861137, 9.772098267560115135805584041789
