| L(s) = 1 | + (1.13 + 1.77i)2-s + (0.180 − 1.72i)3-s + (−1.01 + 2.21i)4-s + (−1.10 + 0.324i)5-s + (3.25 − 1.64i)6-s + (−1.21 + 1.05i)7-s + (−0.918 + 0.132i)8-s + (−2.93 − 0.622i)9-s + (−1.83 − 1.58i)10-s + (−4.10 − 2.63i)11-s + (3.64 + 2.14i)12-s + (2.30 − 2.66i)13-s + (−3.26 − 0.958i)14-s + (0.359 + 1.96i)15-s + (1.91 + 2.21i)16-s + (2.44 + 5.35i)17-s + ⋯ |
| L(s) = 1 | + (0.805 + 1.25i)2-s + (0.104 − 0.994i)3-s + (−0.506 + 1.10i)4-s + (−0.494 + 0.145i)5-s + (1.33 − 0.670i)6-s + (−0.461 + 0.399i)7-s + (−0.324 + 0.0466i)8-s + (−0.978 − 0.207i)9-s + (−0.580 − 0.502i)10-s + (−1.23 − 0.794i)11-s + (1.05 + 0.619i)12-s + (0.640 − 0.738i)13-s + (−0.872 − 0.256i)14-s + (0.0927 + 0.506i)15-s + (0.478 + 0.552i)16-s + (0.593 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09523 + 0.500220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09523 + 0.500220i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.180 + 1.72i)T \) |
| 23 | \( 1 + (2.74 + 3.93i)T \) |
| good | 2 | \( 1 + (-1.13 - 1.77i)T + (-0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (1.10 - 0.324i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.21 - 1.05i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.10 + 2.63i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.30 + 2.66i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.44 - 5.35i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-3.45 - 1.57i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-4.87 + 2.22i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.200 + 1.39i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (0.670 - 2.28i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.848 - 2.89i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.65 - 0.238i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 11.0iT - 47T^{2} \) |
| 53 | \( 1 + (4.00 + 4.62i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.12 - 3.57i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (3.50 - 0.503i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (7.43 + 11.5i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (0.503 + 0.784i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.17 + 4.75i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.834 + 0.722i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-9.84 - 2.89i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.477 - 3.32i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.01 + 10.2i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−14.95009162903977529680800970438, −13.77095046956063073747195886638, −13.08242652297575667185581214446, −12.14071693017409329785732303947, −10.56098621578661012014291869683, −8.238096296538793748853814812134, −7.84108892686103740801594817419, −6.29478022657838396878312128421, −5.58257002099614484902795684159, −3.36161806427539556972414988990,
2.93069180611733879040430304618, 4.17485501230421630422263240797, 5.25139384246629179758380919327, 7.63357010756295092188507146012, 9.481342186995049461316017150843, 10.29781971652030830697674680699, 11.35812545881976125937298353123, 12.20322467185371723896372932673, 13.52695129880856747749491402114, 14.21444220310351994653146776821
