L(s) = 1 | − 5.08·2-s + 2.08·3-s − 6.16·4-s + 41.8·5-s − 10.5·6-s + 193.·8-s − 238.·9-s − 212.·10-s + 72.0·11-s − 12.8·12-s − 632.·13-s + 87.1·15-s − 788.·16-s − 1.97e3·17-s + 1.21e3·18-s − 1.86e3·19-s − 257.·20-s − 366.·22-s + 413.·23-s + 404.·24-s − 1.37e3·25-s + 3.21e3·26-s − 1.00e3·27-s + 731.·29-s − 442.·30-s + 6.12e3·31-s − 2.19e3·32-s + ⋯ |
L(s) = 1 | − 0.898·2-s + 0.133·3-s − 0.192·4-s + 0.748·5-s − 0.120·6-s + 1.07·8-s − 0.982·9-s − 0.672·10-s + 0.179·11-s − 0.0257·12-s − 1.03·13-s + 0.0999·15-s − 0.770·16-s − 1.65·17-s + 0.882·18-s − 1.18·19-s − 0.144·20-s − 0.161·22-s + 0.163·23-s + 0.143·24-s − 0.440·25-s + 0.932·26-s − 0.264·27-s + 0.161·29-s − 0.0898·30-s + 1.14·31-s − 0.379·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
good | 2 | \( 1 + 5.08T + 32T^{2} \) |
| 3 | \( 1 - 2.08T + 243T^{2} \) |
| 5 | \( 1 - 41.8T + 3.12e3T^{2} \) |
| 11 | \( 1 - 72.0T + 1.61e5T^{2} \) |
| 13 | \( 1 + 632.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.97e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.86e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 413.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 731.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.12e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.52e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.14e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.25e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.02e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.05e5T + 8.58e9T^{2} \) |
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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
−13.97707505916677750572864290375, −13.07045261790796160774708673764, −11.37549949434715106644384913737, −10.12561173926461168098254101013, −9.135193729930420440753916975988, −8.180280829456720905351442699079, −6.47594150688748095783940565143, −4.70786807799000360944408023498, −2.21371360549040054329945564287, 0,
2.21371360549040054329945564287, 4.70786807799000360944408023498, 6.47594150688748095783940565143, 8.180280829456720905351442699079, 9.135193729930420440753916975988, 10.12561173926461168098254101013, 11.37549949434715106644384913737, 13.07045261790796160774708673764, 13.97707505916677750572864290375