| L(s) = 1 | + 5.88·2-s + 14.8·3-s + 2.63·4-s + 25·5-s + 87.2·6-s − 140.·7-s − 172.·8-s − 23.3·9-s + 147.·10-s − 121·11-s + 39.0·12-s + 993.·13-s − 825.·14-s + 370.·15-s − 1.10e3·16-s + 2.06e3·17-s − 137.·18-s + 361·19-s + 65.9·20-s − 2.07e3·21-s − 712.·22-s − 1.36e3·23-s − 2.56e3·24-s + 625·25-s + 5.84e3·26-s − 3.94e3·27-s − 369.·28-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.950·3-s + 0.0824·4-s + 0.447·5-s + 0.989·6-s − 1.08·7-s − 0.954·8-s − 0.0959·9-s + 0.465·10-s − 0.301·11-s + 0.0783·12-s + 1.63·13-s − 1.12·14-s + 0.425·15-s − 1.07·16-s + 1.73·17-s − 0.0998·18-s + 0.229·19-s + 0.0368·20-s − 1.02·21-s − 0.313·22-s − 0.538·23-s − 0.907·24-s + 0.200·25-s + 1.69·26-s − 1.04·27-s − 0.0891·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 5.88T + 32T^{2} \) |
| 3 | \( 1 - 14.8T + 243T^{2} \) |
| 7 | \( 1 + 140.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 993.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.06e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.36e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.24e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.05e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.19e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.79e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.22e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.20e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.16e4T + 8.58e9T^{2} \) |
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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
−8.701850824259951527982211887851, −8.164887285147009232213561112321, −6.88050299169822002380805525551, −5.85547864502808538178923850738, −5.56994138247298321157742762701, −4.08743756461220867234104864957, −3.26334819128414106023058733365, −2.99466932296359106030326136260, −1.50954253893343967574795743507, 0,
1.50954253893343967574795743507, 2.99466932296359106030326136260, 3.26334819128414106023058733365, 4.08743756461220867234104864957, 5.56994138247298321157742762701, 5.85547864502808538178923850738, 6.88050299169822002380805525551, 8.164887285147009232213561112321, 8.701850824259951527982211887851
