| L(s) = 1 | − 21.7·2-s + 54.2·3-s + 346.·4-s − 409.·5-s − 1.18e3·6-s + 476.·7-s − 4.76e3·8-s + 761.·9-s + 8.93e3·10-s + 6.86e3·11-s + 1.88e4·12-s − 8.27e3·13-s − 1.03e4·14-s − 2.22e4·15-s + 5.94e4·16-s − 1.61e4·17-s − 1.65e4·18-s + 2.75e4·19-s − 1.42e5·20-s + 2.58e4·21-s − 1.49e5·22-s − 6.91e4·23-s − 2.58e5·24-s + 8.99e4·25-s + 1.80e5·26-s − 7.74e4·27-s + 1.65e5·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 1.16·3-s + 2.70·4-s − 1.46·5-s − 2.23·6-s + 0.525·7-s − 3.28·8-s + 0.348·9-s + 2.82·10-s + 1.55·11-s + 3.14·12-s − 1.04·13-s − 1.01·14-s − 1.70·15-s + 3.62·16-s − 0.795·17-s − 0.670·18-s + 0.920·19-s − 3.97·20-s + 0.609·21-s − 2.99·22-s − 1.18·23-s − 3.81·24-s + 1.15·25-s + 2.01·26-s − 0.756·27-s + 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 + 21.7T + 128T^{2} \) |
| 3 | \( 1 - 54.2T + 2.18e3T^{2} \) |
| 5 | \( 1 + 409.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 476.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.86e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.27e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.61e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.75e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.91e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.11e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.69e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.01e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.13e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 6.24e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.65e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.43e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.53e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.27e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 8.21e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.08e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.74e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.81e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.40e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.78e6T + 8.07e13T^{2} \) |
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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
−14.51771442674692958085902909132, −11.92739954796748580547582393398, −11.38190488523409730115706469783, −9.669148657139648966079214350676, −8.737111072649752227342659734028, −7.892463385712967244748099971997, −7.04321258789678050528309888381, −3.55933564106661000678463628052, −1.84616081283351712840215811817, 0,
1.84616081283351712840215811817, 3.55933564106661000678463628052, 7.04321258789678050528309888381, 7.892463385712967244748099971997, 8.737111072649752227342659734028, 9.669148657139648966079214350676, 11.38190488523409730115706469783, 11.92739954796748580547582393398, 14.51771442674692958085902909132
