| L(s) = 1 | + (1.14 + 0.201i)2-s + (1.10 + 2.78i)3-s + (−2.49 − 0.907i)4-s + (−3.46 − 4.12i)5-s + (0.700 + 3.41i)6-s + (9.89 − 3.60i)7-s + (−6.68 − 3.86i)8-s + (−6.55 + 6.16i)9-s + (−3.12 − 5.41i)10-s + (−7.54 + 8.99i)11-s + (−0.223 − 7.95i)12-s + (1.95 + 11.0i)13-s + (12.0 − 2.12i)14-s + (7.67 − 14.2i)15-s + (1.25 + 1.05i)16-s + (2.73 − 1.57i)17-s + ⋯ |
| L(s) = 1 | + (0.571 + 0.100i)2-s + (0.368 + 0.929i)3-s + (−0.623 − 0.226i)4-s + (−0.692 − 0.824i)5-s + (0.116 + 0.568i)6-s + (1.41 − 0.514i)7-s + (−0.836 − 0.482i)8-s + (−0.728 + 0.684i)9-s + (−0.312 − 0.541i)10-s + (−0.686 + 0.817i)11-s + (−0.0186 − 0.662i)12-s + (0.150 + 0.850i)13-s + (0.860 − 0.151i)14-s + (0.511 − 0.947i)15-s + (0.0786 + 0.0659i)16-s + (0.160 − 0.0929i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.11569 + 0.211812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11569 + 0.211812i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.10 - 2.78i)T \) |
| good | 2 | \( 1 + (-1.14 - 0.201i)T + (3.75 + 1.36i)T^{2} \) |
| 5 | \( 1 + (3.46 + 4.12i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-9.89 + 3.60i)T + (37.5 - 31.4i)T^{2} \) |
| 11 | \( 1 + (7.54 - 8.99i)T + (-21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (-1.95 - 11.0i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (-2.73 + 1.57i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-2.26 + 3.91i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.80 + 18.7i)T + (-405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-24.9 - 4.40i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (20.7 + 7.56i)T + (736. + 617. i)T^{2} \) |
| 37 | \( 1 + (8.82 + 15.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (12.2 - 2.16i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-27.9 - 23.4i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (4.24 + 11.6i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 - 84.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (43.8 + 52.3i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (73.3 - 26.7i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (16.8 + 95.6i)T + (-4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-105. + 60.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (45.5 - 78.9i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (3.54 - 20.0i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-31.1 - 5.48i)T + (6.47e3 + 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-59.7 - 34.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (61.1 + 51.2i)T + (1.63e3 + 9.26e3i)T^{2} \) |
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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
−17.00042772337849543450879926640, −15.72398204618220587288620432950, −14.69312133283075074178114468986, −13.82555964868480975315185158294, −12.27014451453358708409168330567, −10.75178150676235283218372276269, −9.160998548911630437439014342933, −8.000233054041624668843587889173, −4.91526461161042851879146963727, −4.34256022521148823902283242920,
3.17291295981890853850787157365, 5.49214456490261021160277510317, 7.72499551191467243402003880822, 8.529405701842023629514306966778, 11.10947671057623635375213095342, 12.12132147718583582571352224526, 13.44328312849917082649474013361, 14.44804027433038333380694780076, 15.27855974107988806540606453480, 17.59578660582892650565811896463
