| L(s) = 1 | + 4·2-s + 1.29·3-s + 16·4-s − 65.5·5-s + 5.16·6-s − 195.·7-s + 64·8-s − 241.·9-s − 262.·10-s + 309.·11-s + 20.6·12-s − 173.·13-s − 781.·14-s − 84.7·15-s + 256·16-s − 852.·17-s − 965.·18-s + 1.71e3·19-s − 1.04e3·20-s − 252.·21-s + 1.23e3·22-s − 4.59e3·23-s + 82.6·24-s + 1.17e3·25-s − 692.·26-s − 625.·27-s − 3.12e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0828·3-s + 0.5·4-s − 1.17·5-s + 0.0585·6-s − 1.50·7-s + 0.353·8-s − 0.993·9-s − 0.829·10-s + 0.772·11-s + 0.0414·12-s − 0.284·13-s − 1.06·14-s − 0.0971·15-s + 0.250·16-s − 0.715·17-s − 0.702·18-s + 1.08·19-s − 0.586·20-s − 0.124·21-s + 0.546·22-s − 1.81·23-s + 0.0292·24-s + 0.376·25-s − 0.200·26-s − 0.165·27-s − 0.753·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 4T \) |
| 31 | \( 1 - 961T \) |
| good | 3 | \( 1 - 1.29T + 243T^{2} \) |
| 5 | \( 1 + 65.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 309.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 173.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 852.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.71e3T + 2.05e7T^{2} \) |
| 37 | \( 1 + 8.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.34e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.37e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.72e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 178.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.80e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.28e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.51e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.61e4T + 8.58e9T^{2} \) |
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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
−13.51873738458809780133211346402, −12.11276545870694558828264648794, −11.70338527769543751515613993861, −10.06147816545480840107811126991, −8.611998166266742566444613835899, −7.13686187589365638110081421081, −5.96774307107353619865578143588, −4.07251864933611745057777069066, −3.00455380786976297820571817674, 0,
3.00455380786976297820571817674, 4.07251864933611745057777069066, 5.96774307107353619865578143588, 7.13686187589365638110081421081, 8.611998166266742566444613835899, 10.06147816545480840107811126991, 11.70338527769543751515613993861, 12.11276545870694558828264648794, 13.51873738458809780133211346402
