| L(s) = 1 | + (−1.98 + 0.723i)2-s + (1.89 − 1.59i)4-s + (0.465 + 2.64i)5-s + (0.744 + 0.625i)7-s + (−0.508 + 0.880i)8-s + (−2.83 − 4.91i)10-s + (0.0550 − 0.312i)11-s + (1.42 + 0.517i)13-s + (−1.93 − 0.703i)14-s + (−0.487 + 2.76i)16-s + (0.587 + 1.01i)17-s + (−3.11 + 5.38i)19-s + (5.09 + 4.27i)20-s + (0.116 + 0.660i)22-s + (−1.65 + 1.39i)23-s + ⋯ |
| L(s) = 1 | + (−1.40 + 0.511i)2-s + (0.949 − 0.797i)4-s + (0.208 + 1.18i)5-s + (0.281 + 0.236i)7-s + (−0.179 + 0.311i)8-s + (−0.897 − 1.55i)10-s + (0.0165 − 0.0940i)11-s + (0.394 + 0.143i)13-s + (−0.516 − 0.188i)14-s + (−0.121 + 0.691i)16-s + (0.142 + 0.246i)17-s + (−0.713 + 1.23i)19-s + (1.13 + 0.956i)20-s + (0.0248 + 0.140i)22-s + (−0.345 + 0.289i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.289599 + 0.515769i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.289599 + 0.515769i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (1.98 - 0.723i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.465 - 2.64i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.744 - 0.625i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0550 + 0.312i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.42 - 0.517i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.587 - 1.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.11 - 5.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.65 - 1.39i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.13 - 1.50i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (6.64 - 5.57i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.49 - 1.99i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.970 + 5.50i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.89 - 1.59i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + (-0.299 - 1.69i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.777 - 0.652i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (0.804 + 0.292i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (4.79 + 8.31i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.62 + 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 3.83i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.40 + 1.60i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (7.74 - 13.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.963 - 5.46i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.18843084781772792805721668231, −10.87618019205977778088053878188, −10.51238164082780472594764599326, −9.461193379649236785944491135000, −8.500967543610823149591652025440, −7.60186638744036927482592269102, −6.68810760828493221590855640062, −5.79070780828645673995308011542, −3.69609841455787337392146564126, −1.88512261157568814391520633646,
0.77384824577497520871276870787, 2.23079201863606320693805450311, 4.31542853001784878841602107025, 5.58203333462471633087711713584, 7.24209887126731442747328456820, 8.211504919880736715276478736925, 9.028410077693750921903893672654, 9.591753360679519139673214529745, 10.80123148420795025891816797307, 11.41083412610120644312111566760
