| L(s) = 1 | + 3.41·3-s + 2.23·5-s − 0.558i·7-s + 2.66·9-s + (−2.82 + 10.6i)11-s + 18.1i·13-s + 7.63·15-s + 32.0i·17-s − 15.7i·19-s − 1.90i·21-s − 31.7·23-s + 5.00·25-s − 21.6·27-s + 48.3i·29-s + 43.3·31-s + ⋯ |
| L(s) = 1 | + 1.13·3-s + 0.447·5-s − 0.0798i·7-s + 0.296·9-s + (−0.256 + 0.966i)11-s + 1.39i·13-s + 0.509·15-s + 1.88i·17-s − 0.828i·19-s − 0.0908i·21-s − 1.38·23-s + 0.200·25-s − 0.801·27-s + 1.66i·29-s + 1.39·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.603873064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.603873064\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 11 | \( 1 + (2.82 - 10.6i)T \) |
| good | 3 | \( 1 - 3.41T + 9T^{2} \) |
| 7 | \( 1 + 0.558iT - 49T^{2} \) |
| 13 | \( 1 - 18.1iT - 169T^{2} \) |
| 17 | \( 1 - 32.0iT - 289T^{2} \) |
| 19 | \( 1 + 15.7iT - 361T^{2} \) |
| 23 | \( 1 + 31.7T + 529T^{2} \) |
| 29 | \( 1 - 48.3iT - 841T^{2} \) |
| 31 | \( 1 - 43.3T + 961T^{2} \) |
| 37 | \( 1 - 21.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 75.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 8.43T + 2.80e3T^{2} \) |
| 59 | \( 1 - 29.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 64.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 18.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 94.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 0.945iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 73.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 161. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 91.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 33.1T + 9.40e3T^{2} \) |
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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
−10.03582327472225649083727512971, −9.073227133294338390241040554474, −8.652705420881113507650822292576, −7.66408162099066377967033547687, −6.80810311268332749956845151645, −5.86334564125651811823416780812, −4.52111117726785020044754959375, −3.76139597067505044261882000644, −2.40976512992413752774081638968, −1.74489570267849391646692214006,
0.70851446498123375639946076141, 2.51778596494626478776395782314, 2.94781371127871812764402121420, 4.17362966126054230766098932905, 5.53737960409577755835963229996, 6.12119179875490517250386759308, 7.65875242564350356011959772316, 8.022329133160864681693605682286, 8.863768909654952061538595653005, 9.774091510532912913449963947252
