| L(s) = 1 | + (0.642 + 0.233i)2-s + (−1.17 − 0.984i)4-s + (−0.181 + 1.03i)5-s + (−0.0923 + 0.0775i)7-s + (−1.20 − 2.09i)8-s + (−0.358 + 0.620i)10-s + (0.943 + 5.35i)11-s + (4.29 − 1.56i)13-s + (−0.0775 + 0.0282i)14-s + (0.245 + 1.39i)16-s + (2.38 − 4.13i)17-s + (0.294 + 0.509i)19-s + (1.22 − 1.03i)20-s + (−0.645 + 3.66i)22-s + (5.97 + 5.01i)23-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.165i)2-s + (−0.586 − 0.492i)4-s + (−0.0813 + 0.461i)5-s + (−0.0349 + 0.0293i)7-s + (−0.427 − 0.739i)8-s + (−0.113 + 0.196i)10-s + (0.284 + 1.61i)11-s + (1.19 − 0.433i)13-s + (−0.0207 + 0.00754i)14-s + (0.0612 + 0.347i)16-s + (0.579 − 1.00i)17-s + (0.0675 + 0.116i)19-s + (0.275 − 0.230i)20-s + (−0.137 + 0.780i)22-s + (1.24 + 1.04i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67641 + 0.346152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67641 + 0.346152i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.642 - 0.233i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.181 - 1.03i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.0923 - 0.0775i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.943 - 5.35i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.29 + 1.56i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.38 + 4.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.294 - 0.509i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.97 - 5.01i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.76 - 1.73i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (6.70 + 5.63i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.10 + 2.58i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.226 + 1.28i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (1.85 - 1.55i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 3.04T + 53T^{2} \) |
| 59 | \( 1 + (-0.00762 + 0.0432i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (7.82 - 6.56i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.74 + 0.636i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.25 - 5.63i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.11 + 10.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.659 + 0.240i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.36 - 2.31i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.42 - 5.93i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.56 + 8.89i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.44177879124972250800600664131, −9.470067786587771275196411812823, −9.055521816802536017927216497474, −7.59554158341756734626616007604, −6.93898092166808452378633678091, −5.86921689389846636095803285666, −5.04321989131413169111655661333, −4.08507493492183438113191013566, −3.03008764858125470099968705371, −1.24916648219517238400517734437,
1.01405511610628176487314024415, 3.03470944070440703565439042258, 3.76586560371878806510020085784, 4.76748269683438096516517054443, 5.76932299113440177944282020243, 6.61938785623436595424691311510, 8.126751217246614469060189054561, 8.633072560420172535502723731001, 9.116702482781640699854528422401, 10.60248556635714366924830146656
