L(s) = 1 | + (−0.5 − 0.866i)3-s + 5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + 21-s + (0.5 + 0.866i)23-s + 25-s + 27-s + (0.5 + 0.866i)29-s + 31-s + (0.5 − 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + 5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + 21-s + (0.5 + 0.866i)23-s + 25-s + 27-s + (0.5 + 0.866i)29-s + 31-s + (0.5 − 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.469759334 + 0.4039388098i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.469759334 + 0.4039388098i\) |
\(L(1)\) |
\(\approx\) |
\(1.067058220 + 0.01368435880i\) |
\(L(1)\) |
\(\approx\) |
\(1.067058220 + 0.01368435880i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + T \) |
| 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 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 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
−29.188130169917287374168489058598, −28.6644305548210518559220803081, −27.07977883387228510840204991317, −26.63580938991335027383718744034, −25.40354549531544962266508782535, −24.28042525776768898978638648477, −22.80429082056762554636361240684, −22.28369218002022411082905718136, −21.06441087542475272451805819903, −20.361380452743022436234511944017, −18.831493384114091035277031247721, −17.44727974473013530611299999523, −16.76479999640116125118336249946, −15.84608787519206771703447491607, −14.28269651330522567465458281918, −13.5237836784220866875259898522, −11.932121411986731881396440339210, −10.63933945616126462579863663786, −9.88069692202636063957018295517, −8.79952005128423266839034177917, −6.75799398717826870672030393544, −5.79630429585982221964808173847, −4.43254381036835566955231921860, −3.04594545321128796953490673115, −0.76826862225171870991262097020,
1.46424222172001278964961693612, 2.66866313440044051002788680370, 5.02861742291624996869061426539, 6.14817623158702127844767004656, 7.00974469004392563646352406122, 8.72612109800542187876624580358, 9.82831371350558415696540602739, 11.285543531033590032713819872553, 12.51332070105726470030576376978, 13.20196219232550764039629956206, 14.46904206086167308699540872693, 15.85080866286412862827650298906, 17.35494160198535760198444363604, 17.76952127247303782544422490971, 18.98312366500148433006473683608, 19.96417260444996972064231645046, 21.60365818740086743052357841656, 22.2571868728314383588815670824, 23.3700121920613906675248915691, 24.68443418638207155002417063559, 25.22414246240529542316063946072, 26.19511223335934151386444880409, 28.1069934813781177979727850002, 28.514963339777991679736811712851, 29.546377427231459564965984488850