L(s) = 1 | + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (0.309 + 0.951i)5-s + (0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.978 − 0.207i)13-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (0.104 + 0.994i)19-s + (0.669 + 0.743i)20-s − 23-s + (−0.809 + 0.587i)25-s + (0.913 − 0.406i)26-s + (−0.913 + 0.406i)29-s + (−0.669 − 0.743i)31-s + (0.5 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (0.309 + 0.951i)5-s + (0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.978 − 0.207i)13-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (0.104 + 0.994i)19-s + (0.669 + 0.743i)20-s − 23-s + (−0.809 + 0.587i)25-s + (0.913 − 0.406i)26-s + (−0.913 + 0.406i)29-s + (−0.669 − 0.743i)31-s + (0.5 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.022998377 + 0.1029273382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.022998377 + 0.1029273382i\) |
\(L(1)\) |
\(\approx\) |
\(2.092414459 + 0.01412307312i\) |
\(L(1)\) |
\(\approx\) |
\(2.092414459 + 0.01412307312i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (0.978 - 0.207i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (0.104 + 0.994i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.913 + 0.406i)T \) |
| 31 | \( 1 + (-0.669 - 0.743i)T \) |
| 37 | \( 1 + (0.913 - 0.406i)T \) |
| 41 | \( 1 + (0.913 + 0.406i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.913 + 0.406i)T \) |
| 53 | \( 1 + (0.978 - 0.207i)T \) |
| 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 | \( 1 + (0.104 - 0.994i)T \) |
| 79 | \( 1 + (-0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.978 - 0.207i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.669 - 0.743i)T \) |
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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
−22.710917408385099167681300744597, −21.67162533472938772382922561273, −21.2997791256284645748835845359, −20.30020957272922448954449807164, −19.86573965044594058052006916577, −18.59491412909013697673988700257, −17.479899247583700074537422385202, −16.72996858529249836089222559560, −16.03188897646051040880592728833, −15.289944793512266587746248285775, −14.23747336677008741633751372019, −13.483087674298363525614015453411, −12.855691735064134097703335926556, −12.03543934108892254731401243620, −11.18739926371319658928811706126, −10.17073513798444723821840341874, −8.96347631881306856656069800219, −8.171459045335312379953795222986, −7.14721403380670036983467399215, −5.97382034267103115768697113826, −5.498594956963511650717674111, −4.3415615508348533854929342759, −3.67134122803466076938235235704, −2.2960465002294676988188146228, −1.281824604291389139352147903174,
1.419741064529414719755103936425, 2.49663381867035291574258390792, 3.42844946200675726475569156912, 4.17149731501826557258116366085, 5.72808338361299939765374234938, 5.94406316363223310918327090341, 7.17720109027219419148095158104, 7.88492392473792731074560559509, 9.49374053336811931683588406504, 10.287019292103168457619878193657, 11.10987214480658530549465987304, 11.7810214356064543024774201184, 12.85092280040531832157979783744, 13.63970853985631227782685755666, 14.40662301271880568396563348557, 14.94507429310848716767082125687, 16.03424674563887768192351856134, 16.619755542760800025896719682614, 18.1282520419375653569543616627, 18.53360119162537881274590454614, 19.55802216549589058004002974066, 20.53061327554969485579463450167, 21.0994048881274584287351261428, 22.01481298940262684214593979403, 22.673649370079129169518608570129