L(s) = 1 | + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s − 53-s + (0.5 + 0.866i)59-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s − 53-s + (0.5 + 0.866i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08478323559 + 0.4808296226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08478323559 + 0.4808296226i\) |
\(L(1)\) |
\(\approx\) |
\(0.7988324569 + 0.1408557152i\) |
\(L(1)\) |
\(\approx\) |
\(0.7988324569 + 0.1408557152i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + T \) |
| 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
−26.576041018024390109539947702327, −25.742198680402967679227680981559, −24.8747659097786287075334677222, −23.62860733978518762047928990377, −22.870811428529201163897847700083, −22.03780853303737108437670036865, −20.63269742018073258746592229234, −20.0053993883633338216994786922, −19.0342556856564974758541191467, −17.64784832315501990781673800415, −17.09388332101134560705811282642, −15.766876553576468120000113133060, −14.96068047031785287236709961494, −13.58055051266230690330255996100, −12.93551838266130740775020989541, −11.61641561981770368198516276707, −10.468048540135157515659885054250, −9.60708230976048022713222515821, −8.26725489011218214527040361026, −7.088258551572188526337351203, −6.122493576298045876595819528, −4.53053707662123979157856388912, −3.53370181755788688118107496246, −1.86310405224292028472763200672, −0.16397356514340721383071475628,
1.81263338074527618888872419919, 3.205884482504983972106960170806, 4.520566899505271874868067696847, 6.04386621655767266651217941899, 6.72544624423465379238770853777, 8.58867019905235862496014227127, 9.00884084848137959601367515413, 10.531050582491404991413013743, 11.568438768182544050799621497170, 12.57559554895601309269002250059, 13.6721025743400107090446660419, 14.71600711785655947296565670013, 15.87396599901817723316239051464, 16.59109130592089197806180462087, 17.8878678587834677466910754327, 18.914207348125614932957405010465, 19.54617123816639698301163655354, 20.92218415878567481242816255630, 21.82403316264162670571406837516, 22.531269835120791810700985771141, 23.844358075256018180414021159964, 24.60058636052511889580135010143, 25.65227179001161090697899407043, 26.4645383218042554060981632316, 27.57859346429966183339175390428