| L(s) = 1 | − 8.57·2-s − 9·3-s + 41.5·4-s + 77.1·6-s − 178.·7-s − 81.7·8-s + 81·9-s + 121·11-s − 373.·12-s + 361.·13-s + 1.53e3·14-s − 627.·16-s − 934.·17-s − 694.·18-s + 753.·19-s + 1.60e3·21-s − 1.03e3·22-s − 3.23e3·23-s + 735.·24-s − 3.10e3·26-s − 729·27-s − 7.41e3·28-s + 2.60e3·29-s + 662.·31-s + 8.00e3·32-s − 1.08e3·33-s + 8.00e3·34-s + ⋯ |
| L(s) = 1 | − 1.51·2-s − 0.577·3-s + 1.29·4-s + 0.875·6-s − 1.37·7-s − 0.451·8-s + 0.333·9-s + 0.301·11-s − 0.749·12-s + 0.594·13-s + 2.08·14-s − 0.613·16-s − 0.783·17-s − 0.505·18-s + 0.479·19-s + 0.794·21-s − 0.457·22-s − 1.27·23-s + 0.260·24-s − 0.900·26-s − 0.192·27-s − 1.78·28-s + 0.575·29-s + 0.123·31-s + 1.38·32-s − 0.174·33-s + 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 8.57T + 32T^{2} \) |
| 7 | \( 1 + 178.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 361.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 934.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 753.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 662.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.29e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.08e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.75e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.29e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.36e5T + 8.58e9T^{2} \) |
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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
−9.119311963877995874049552096386, −8.412970669236474084381010722675, −7.39515500378832856808768694003, −6.53181299650537164032019953421, −6.06653297064713990704581704129, −4.53587008193812360321373260284, −3.34353758442410332535888394930, −2.02476971779981580094426492904, −0.836058620815318938427685584803, 0,
0.836058620815318938427685584803, 2.02476971779981580094426492904, 3.34353758442410332535888394930, 4.53587008193812360321373260284, 6.06653297064713990704581704129, 6.53181299650537164032019953421, 7.39515500378832856808768694003, 8.412970669236474084381010722675, 9.119311963877995874049552096386
