L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (0.623 + 0.781i)8-s + (0.988 + 0.149i)10-s + (0.955 + 0.294i)11-s + (−0.955 − 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 + 0.997i)26-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (0.623 + 0.781i)8-s + (0.988 + 0.149i)10-s + (0.955 + 0.294i)11-s + (−0.955 − 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 + 0.997i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.171668392 + 0.2652311678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.171668392 + 0.2652311678i\) |
\(L(1)\) |
\(\approx\) |
\(0.8236059955 - 0.1628446331i\) |
\(L(1)\) |
\(\approx\) |
\(0.8236059955 - 0.1628446331i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (-0.365 + 0.930i)T \) |
| 11 | \( 1 + (0.955 + 0.294i)T \) |
| 13 | \( 1 + (-0.955 - 0.294i)T \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.826 + 0.563i)T \) |
| 29 | \( 1 + (0.0747 + 0.997i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.826 - 0.563i)T \) |
| 41 | \( 1 + (0.988 - 0.149i)T \) |
| 43 | \( 1 + (-0.988 - 0.149i)T \) |
| 47 | \( 1 + (0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.826 + 0.563i)T \) |
| 59 | \( 1 + (-0.623 + 0.781i)T \) |
| 61 | \( 1 + (0.900 + 0.433i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.955 + 0.294i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.955 + 0.294i)T \) |
| 89 | \( 1 + (0.733 + 0.680i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−23.96461061487898205639431849043, −23.179836190937179201918432104461, −22.235389496944874832564181996737, −21.38947642053249823860705078744, −20.09899592333687146412184806162, −19.41169460158783978670341562174, −18.60718593637506908419159658693, −17.301114231755706345154505068141, −16.84162647429881282073925886008, −16.19143033615863067745957480615, −15.01266824074375237490755559265, −14.477273785943353288975766127606, −13.30494321238271905540826360225, −12.49386947610494740109484644154, −11.506880299354229251044766461495, −10.05864804520378847306437730415, −9.23781248174903600251493686795, −8.42983382485464657483874834838, −7.60194344953457226660085390422, −6.52164619131666442118702078675, −5.54051049096106545237790559602, −4.54245811146534952473550626934, −3.721981631443335156594945920698, −1.62948127710015633980376514849, −0.45638089052305783828282170208,
0.93447387796304048642602520284, 2.41663305939565949873871710803, 3.17686713387171371043163560207, 4.26190741564497378525294470132, 5.33066422604433183204051504826, 7.01870449995093988655105332233, 7.53126259548067481998839059797, 9.0462948948347522820608238457, 9.62537058143164508506342060728, 10.74414555537481321958903361517, 11.45805636656465205234585624973, 12.172712915407320799054337391466, 13.242808205551092002255686594959, 14.304240138383523507235625883829, 14.88097226799619118483598258039, 16.192510068166016385448211528153, 17.349218605992555310908040665878, 17.99592273968678874680437525296, 18.84709119798040974405938330051, 19.80358216927837411510689707236, 20.08985644454829272027706854244, 21.49978878497838326308564610877, 22.15653250163710352689834415430, 22.71660520920660422852006376985, 23.60881432996894840881390548817