| L(s) = 1 | + (−0.145 − 0.0388i)2-s + (0.490 − 5.17i)3-s + (−6.90 − 3.98i)4-s + (−0.272 + 0.731i)6-s + (4.62 − 17.2i)7-s + (1.69 + 1.69i)8-s + (−26.5 − 5.07i)9-s + (21.0 − 12.1i)11-s + (−24.0 + 33.7i)12-s + (−13.8 − 51.6i)13-s + (−1.34 + 2.32i)14-s + (31.7 + 54.9i)16-s + (−62.6 + 62.6i)17-s + (3.65 + 1.76i)18-s − 42.2i·19-s + ⋯ |
| L(s) = 1 | + (−0.0513 − 0.0137i)2-s + (0.0943 − 0.995i)3-s + (−0.863 − 0.498i)4-s + (−0.0185 + 0.0497i)6-s + (0.249 − 0.931i)7-s + (0.0750 + 0.0750i)8-s + (−0.982 − 0.187i)9-s + (0.577 − 0.333i)11-s + (−0.577 + 0.812i)12-s + (−0.295 − 1.10i)13-s + (−0.0256 + 0.0443i)14-s + (0.495 + 0.858i)16-s + (−0.893 + 0.893i)17-s + (0.0478 + 0.0231i)18-s − 0.510i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.197536 + 0.676627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.197536 + 0.676627i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.490 + 5.17i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.145 + 0.0388i)T + (6.92 + 4i)T^{2} \) |
| 7 | \( 1 + (-4.62 + 17.2i)T + (-297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-21.0 + 12.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (13.8 + 51.6i)T + (-1.90e3 + 1.09e3i)T^{2} \) |
| 17 | \( 1 + (62.6 - 62.6i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 42.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (43.0 - 11.5i)T + (1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (-94.1 - 163. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (123. - 213. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (127. + 127. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + (32.7 + 18.9i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-131. - 35.1i)T + (6.88e4 + 3.97e4i)T^{2} \) |
| 47 | \( 1 + (-266. - 71.4i)T + (8.99e4 + 5.19e4i)T^{2} \) |
| 53 | \( 1 + (509. + 509. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-218. + 378. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (438. + 759. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (346. - 92.9i)T + (2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 + 534. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (579. - 579. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (522. - 301. i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-15.8 + 58.9i)T + (-4.95e5 - 2.85e5i)T^{2} \) |
| 89 | \( 1 - 858.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-348. + 1.30e3i)T + (-7.90e5 - 4.56e5i)T^{2} \) |
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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
−11.06664813689568087538521911362, −10.41366725618709232612295311920, −9.034939884393310976028250138017, −8.295356124976864266397506969997, −7.19790203337447281353657815325, −6.12112139194217595707623293438, −4.91673551718607512986845286214, −3.50623602882896884635512131873, −1.52421345171607031115860432961, −0.30719957504842860207037929650,
2.48291685550772862129614844581, 4.05966598048702628102947097920, 4.72529353588312408182016508773, 5.99668663814429693527245328098, 7.61469673659434662597743699918, 8.967262790805910443857534313994, 9.107453989677747235918273434404, 10.18466006371939729262868386227, 11.67983356511288885115873357143, 12.00117172404197464792525002069
