| L(s) = 1 | − 9i·5-s − 5·7-s + 117i·11-s − 34·13-s − 450i·17-s + 64·19-s + 612i·23-s + 544·25-s − 1.06e3i·29-s + 697·31-s + 45i·35-s − 748·37-s − 684i·41-s − 2.61e3·43-s − 2.64e3i·47-s + ⋯ |
| L(s) = 1 | − 0.359i·5-s − 0.102·7-s + 0.966i·11-s − 0.201·13-s − 1.55i·17-s + 0.177·19-s + 1.15i·23-s + 0.870·25-s − 1.26i·29-s + 0.725·31-s + 0.0367i·35-s − 0.546·37-s − 0.406i·41-s − 1.41·43-s − 1.19i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.449193623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449193623\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 9iT - 625T^{2} \) |
| 7 | \( 1 + 5T + 2.40e3T^{2} \) |
| 11 | \( 1 - 117iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 34T + 2.85e4T^{2} \) |
| 17 | \( 1 + 450iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 64T + 1.30e5T^{2} \) |
| 23 | \( 1 - 612iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.06e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 697T + 9.23e5T^{2} \) |
| 37 | \( 1 + 748T + 1.87e6T^{2} \) |
| 41 | \( 1 + 684iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.61e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.64e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.07e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.81e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.40e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.21e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 6.57e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.51e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.50e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 5.48e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.87e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.05e4T + 8.85e7T^{2} \) |
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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
−10.01964312669637317859716684862, −9.596634121700847561802868319186, −8.519312219571044341831132872063, −7.45606025443198245414961035692, −6.72006375706422888546555563664, −5.31887974141267104679505997299, −4.61394359760580232699193384008, −3.22514507476873214906039032901, −1.92062792018800602581848058174, −0.42989797161675293592963957028,
1.17940573978854530401552574712, 2.72800916874921561164442697910, 3.74045434850525621541896381032, 5.02620518126748071656722574306, 6.18347007224074524320106045469, 6.88392018319206240868966792698, 8.232791560014739302147420563550, 8.734227492320803443803696480487, 10.07820378319382933309568587270, 10.68918115665637423283838966369
