| L(s) = 1 | + (−2.02 − 0.563i)2-s + (2.07 + 1.25i)4-s + (1.27 − 0.0493i)5-s + (4.39 + 0.687i)7-s + (−0.606 − 0.642i)8-s + (−2.60 − 0.616i)10-s + (−0.0413 − 0.302i)11-s + (1.95 − 2.84i)13-s + (−8.52 − 3.87i)14-s + (−1.39 − 2.63i)16-s + (6.95 − 3.49i)17-s + (0.466 + 8.01i)19-s + (2.69 + 1.48i)20-s + (−0.0869 + 0.636i)22-s + (−1.39 + 3.61i)23-s + ⋯ |
| L(s) = 1 | + (−1.43 − 0.398i)2-s + (1.03 + 0.625i)4-s + (0.568 − 0.0220i)5-s + (1.66 + 0.260i)7-s + (−0.214 − 0.227i)8-s + (−0.822 − 0.194i)10-s + (−0.0124 − 0.0913i)11-s + (0.540 − 0.788i)13-s + (−2.27 − 1.03i)14-s + (−0.347 − 0.659i)16-s + (1.68 − 0.846i)17-s + (0.107 + 1.83i)19-s + (0.602 + 0.332i)20-s + (−0.0185 + 0.135i)22-s + (−0.291 + 0.754i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03021 - 0.171057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03021 - 0.171057i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.02 + 0.563i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-1.27 + 0.0493i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-4.39 - 0.687i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.0413 + 0.302i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-1.95 + 2.84i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-6.95 + 3.49i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.466 - 8.01i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (1.39 - 3.61i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (2.53 + 1.81i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (2.85 - 3.27i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-1.50 - 2.02i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.728 + 2.82i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (6.63 + 6.02i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.229 - 0.263i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (0.771 - 4.37i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (2.57 - 1.05i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-10.2 + 6.16i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-5.48 + 3.91i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-0.934 - 3.12i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (4.31 - 1.02i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-9.02 + 8.85i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-0.101 - 0.393i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (2.03 - 6.80i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (11.4 + 0.445i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.13986127426299067099686824282, −9.673240819946576336182530429348, −8.564260155668320807511637330071, −7.942904416399427013003060200608, −7.50509683235415233356415822679, −5.72220824844852120298096682444, −5.23306483433401731732941322175, −3.50833503466056064965060274301, −1.96759953083089134681940810592, −1.24054350095679281046281993105,
1.16048589962663771562829384342, 2.06647189967051613655908338414, 4.07048913132359727496482800459, 5.18693427164006380176493877269, 6.28750167333261293741715695299, 7.27731716976423759083220367926, 8.013086906340555701791110739362, 8.627392512597548189200135787889, 9.497391741310006659470433959572, 10.22742748127548300161624838994
