L(s) = 1 | + (−0.978 + 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.669 + 0.743i)5-s + (−0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (0.913 − 0.406i)18-s + (−0.913 − 0.406i)19-s + (−0.309 + 0.951i)20-s + ⋯ |
L(s) = 1 | + (−0.978 + 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.669 + 0.743i)5-s + (−0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (0.913 − 0.406i)18-s + (−0.913 − 0.406i)19-s + (−0.309 + 0.951i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1151294862 + 0.2393518603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1151294862 + 0.2393518603i\) |
\(L(1)\) |
\(\approx\) |
\(0.4163337123 + 0.2930042850i\) |
\(L(1)\) |
\(\approx\) |
\(0.4163337123 + 0.2930042850i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.978 + 0.207i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (-0.669 + 0.743i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.978 + 0.207i)T \) |
| 19 | \( 1 + (-0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.669 - 0.743i)T \) |
| 37 | \( 1 + (-0.104 + 0.994i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.913 - 0.406i)T \) |
| 53 | \( 1 + (0.669 + 0.743i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (-0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.913 + 0.406i)T \) |
| 79 | \( 1 + (-0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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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
−29.98477089830405885374150233153, −29.32500934469724953555833412130, −28.009847560651749131602559006685, −27.422379125901819680294003206916, −25.87257146249783772340919687025, −25.02025033940084872183394979925, −24.1028760660918591707044247605, −23.02483407273507932829057173517, −21.11807003492511941507130367933, −19.99098481277878155356646500976, −19.378848354933436454134534477394, −18.230509718397648101674782944648, −17.19745133609336529148914605076, −16.15969993241783000134310447664, −14.73737554947154126119879663085, −12.82508133774253752710043451296, −12.2067221991621554213436275569, −10.94219587603111149695659479081, −9.281947150487250093204122569404, −8.07367609385282539720023899955, −7.42081899179604931759192157120, −5.72749105879624206628640543294, −3.32781154291160252868423602809, −1.57656629180911972297913377485, −0.1723106175390166654072120125,
2.54180706652128659531365323953, 4.13023738987600501882676080644, 6.05018598536805890984049220299, 7.48370415622714958129309334044, 8.716998706615419695995020535211, 9.93776023597150132771095085249, 10.9010291535842373775791005266, 11.8833287636447174090420440532, 14.41497923742296551430684475224, 15.102285681231770018981871289330, 16.244987272577396810237675201176, 17.091857387319626984340266293809, 18.64949369748002627808749641023, 19.45120460443461052727469973693, 20.62217673033795086002125674035, 21.74170608501481308397777562504, 23.05637418238028760706132997749, 24.23703166289592352321642089298, 25.97562911304579747886301244422, 26.18570263733253070329129896992, 27.44974489474998936928093201073, 27.98482079520296381984303566631, 29.31545233808577118241199399070, 30.56001036603616863009962130670, 31.83755547201695287710628229696