L(s) = 1 | + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.222 + 0.974i)5-s + (−0.222 + 0.974i)8-s + (−0.733 + 0.680i)10-s + (−0.900 + 0.433i)11-s + (0.826 + 0.563i)13-s + (−0.733 + 0.680i)16-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.988 + 0.149i)20-s + (−0.988 − 0.149i)22-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.365 + 0.930i)26-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.222 + 0.974i)5-s + (−0.222 + 0.974i)8-s + (−0.733 + 0.680i)10-s + (−0.900 + 0.433i)11-s + (0.826 + 0.563i)13-s + (−0.733 + 0.680i)16-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.988 + 0.149i)20-s + (−0.988 − 0.149i)22-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.365 + 0.930i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5320169930 + 1.703194210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5320169930 + 1.703194210i\) |
\(L(1)\) |
\(\approx\) |
\(1.130240513 + 0.9541029694i\) |
\(L(1)\) |
\(\approx\) |
\(1.130240513 + 0.9541029694i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.826 + 0.563i)T \) |
| 5 | \( 1 + (-0.222 + 0.974i)T \) |
| 11 | \( 1 + (-0.900 + 0.433i)T \) |
| 13 | \( 1 + (0.826 + 0.563i)T \) |
| 17 | \( 1 + (0.365 - 0.930i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.623 - 0.781i)T \) |
| 29 | \( 1 + (-0.988 + 0.149i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.988 + 0.149i)T \) |
| 41 | \( 1 + (0.955 - 0.294i)T \) |
| 43 | \( 1 + (0.955 + 0.294i)T \) |
| 47 | \( 1 + (0.826 + 0.563i)T \) |
| 53 | \( 1 + (-0.988 - 0.149i)T \) |
| 59 | \( 1 + (0.955 + 0.294i)T \) |
| 61 | \( 1 + (0.365 - 0.930i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (0.0747 + 0.997i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.826 - 0.563i)T \) |
| 89 | \( 1 + (0.826 - 0.563i)T \) |
| 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
−23.785137570986666866977927732283, −22.95066068938832181856872353486, −21.87242778195535172262862249937, −20.9758929027031597816860790367, −20.62792066213361807885738689019, −19.51962393812600294336393624919, −18.917438440080809121094848927304, −17.69312707085949441303315608531, −16.590126176642272294398796295831, −15.62106396978982087019208661350, −15.11917822632154553323285235749, −13.71600546141059355863377541207, −13.04656622009586866274495608588, −12.54209757993314583001378861243, −11.267886669820093575901178885847, −10.72665199948546545736090636111, −9.48490460066387991784008081379, −8.514404776065617545313872155776, −7.41715213718849800027315814580, −5.8749192210846043572338151159, −5.38142716040848481778848352636, −4.20702395208300362840575727036, −3.345208487381529043468530056292, −2.014541558450534983085538249349, −0.75753461860332292186942585310,
2.136591276069495336200963682591, 3.1477117927420967444773145241, 4.075410018459206429751092162372, 5.23142850973583578468470362003, 6.238351339708286990853936173688, 7.13657868227454529832048855156, 7.83296024477246806955611594614, 9.01696177884901658446872191139, 10.48983623971700310595970788068, 11.1779224373442346178907469710, 12.2391215155092200796049013830, 13.088855741114801625245525966895, 14.17816868746615794324981489909, 14.62266172059453785288827202682, 15.73769968363557384984917156430, 16.21606207031260246543822331478, 17.43459207732756527176211989944, 18.37808451933637384757685733170, 19.03155315389305005674104159506, 20.60684944460756526588010152596, 20.960831373836592936056445348568, 22.0761515984871844974850978192, 22.93575451208693656588263506201, 23.28194037349320718425712924727, 24.24851503878615296667427559668