L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)11-s + (−0.939 + 0.342i)13-s + (−0.939 + 0.342i)14-s + (0.173 + 0.984i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.766 − 0.642i)20-s + (0.173 − 0.984i)22-s + (0.766 + 0.642i)23-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)11-s + (−0.939 + 0.342i)13-s + (−0.939 + 0.342i)14-s + (0.173 + 0.984i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.766 − 0.642i)20-s + (0.173 − 0.984i)22-s + (0.766 + 0.642i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4904161974 - 0.1948298888i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4904161974 - 0.1948298888i\) |
\(L(1)\) |
\(\approx\) |
\(0.6708527834 - 0.1803680129i\) |
\(L(1)\) |
\(\approx\) |
\(0.6708527834 - 0.1803680129i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.173 - 0.984i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (0.173 + 0.984i)T \) |
| 13 | \( 1 + (-0.939 + 0.342i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 + (0.766 + 0.642i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.939 + 0.342i)T \) |
| 43 | \( 1 + (0.173 + 0.984i)T \) |
| 47 | \( 1 + (0.766 - 0.642i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.939 - 0.342i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.173 + 0.984i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−37.58372954182998689031153394258, −37.01477588187186283406394188866, −35.27352116954896961096017102697, −34.29014437904429629398873340305, −33.53656496223680104973264036537, −31.755868599039456540972199212770, −30.064223929893360388995265648033, −29.06920731570446233616621258167, −27.393782537926762119804255960977, −26.73081007126011074245113306014, −25.204475213023365565372891903, −24.26169998724819837558081461783, −22.415363267806994461457555754775, −20.91543964275324839248893731515, −19.14296493059143658470160211392, −18.286083298559526939498895729894, −16.99631143531635371190370315198, −15.291734837449843918437260545636, −14.28854830331122535283861236746, −11.65396679535830822065831219641, −10.471836307563585828664309239382, −8.81169104513031416092090035974, −7.290203438280219908468402759274, −5.69563784338302977426148653043, −2.4647645825156502465750364336,
1.74993694859259126172085783628, 4.55007778091148975896620945674, 7.16060343511994922697492656148, 8.64039580907036702230177424168, 10.00752396427758668036330038709, 11.62750766701912580283362925472, 13.02402124402460833174722300178, 15.19147760380354340021567674139, 17.03910479009999802770534189648, 17.49383395912093936008620550323, 19.50458733471328116879835740164, 20.451087671551300633152709188393, 21.59841535091784604896837017347, 23.84061236803992608271763504131, 24.95012017012848931506399938462, 26.37212973297575410829294073475, 27.646524949936342287729738193344, 28.55384907944317333973465270055, 29.867208497144359344107460029628, 31.10397322491896607666635155264, 32.906079767968009297468755051336, 34.04395283299891854674509643110, 35.56886624754737798978069059418, 36.48437316178819957089312997132, 37.29804107595635392617108100640