L(s) = 1 | + (−0.994 − 0.104i)2-s + (−0.207 + 0.978i)3-s + (0.978 + 0.207i)4-s + (0.309 − 0.951i)6-s + (−0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)12-s + (0.587 + 0.809i)13-s + (0.913 + 0.406i)16-s + (−0.743 + 0.669i)17-s + (0.866 + 0.5i)18-s + (0.978 − 0.207i)19-s + (−0.951 + 0.309i)22-s + (0.994 + 0.104i)23-s + (0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.104i)2-s + (−0.207 + 0.978i)3-s + (0.978 + 0.207i)4-s + (0.309 − 0.951i)6-s + (−0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)12-s + (0.587 + 0.809i)13-s + (0.913 + 0.406i)16-s + (−0.743 + 0.669i)17-s + (0.866 + 0.5i)18-s + (0.978 − 0.207i)19-s + (−0.951 + 0.309i)22-s + (0.994 + 0.104i)23-s + (0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5733012994 + 0.7895325222i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5733012994 + 0.7895325222i\) |
\(L(1)\) |
\(\approx\) |
\(0.6538402016 + 0.2729548557i\) |
\(L(1)\) |
\(\approx\) |
\(0.6538402016 + 0.2729548557i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.994 - 0.104i)T \) |
| 3 | \( 1 + (-0.207 + 0.978i)T \) |
| 11 | \( 1 + (0.913 - 0.406i)T \) |
| 13 | \( 1 + (0.587 + 0.809i)T \) |
| 17 | \( 1 + (-0.743 + 0.669i)T \) |
| 19 | \( 1 + (0.978 - 0.207i)T \) |
| 23 | \( 1 + (0.994 + 0.104i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.406 + 0.913i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.743 + 0.669i)T \) |
| 53 | \( 1 + (-0.207 + 0.978i)T \) |
| 59 | \( 1 + (0.104 + 0.994i)T \) |
| 61 | \( 1 + (-0.104 + 0.994i)T \) |
| 67 | \( 1 + (-0.743 + 0.669i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.406 - 0.913i)T \) |
| 79 | \( 1 + (-0.669 + 0.743i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.104 - 0.994i)T \) |
| 97 | \( 1 + (-0.951 + 0.309i)T \) |
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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
−26.99870631494576727101090265809, −25.82557612114716902230038938638, −24.925967352705712666385585934295, −24.50242485361902292618568636680, −23.19926605172500067603022283552, −22.32176831536412255873897516132, −20.57985082990114169404296325278, −19.962434370828132019657005149252, −18.94156180162392253882177789538, −18.06272002391845105683569133783, −17.41010874190025506219541893517, −16.39166350442334939009288598241, −15.21879079688749626187589439443, −14.01014603927019301954817612033, −12.73660018378259487161593022154, −11.690350626197972579152749172106, −10.88548436698463232434626840592, −9.45029897096348192362579982255, −8.472967644778310111254825486496, −7.34332001000939466304517548146, −6.58889437703317213086768496712, −5.38155953138030833928647791166, −3.1137282136515459332946258416, −1.70667332592244000667296359018, −0.57473692018332749159404291965,
1.22387695487708211814060442558, 3.023380955802214981236883622957, 4.21965844127345439613638756717, 5.88942098009769029414193379424, 6.90095003790407169905038855639, 8.585419352478314368409740010949, 9.14937081077560526840638322084, 10.24910407590878160080513199978, 11.27398157332485182911024633724, 11.891007888765538495332583777612, 13.73648349812622950987462492921, 15.058694343933003726761480735684, 15.86074914727926670226895293514, 16.84381587141367738415031727305, 17.45198316458614379310065438175, 18.776635981397684055931233085812, 19.715154364826232927697356495391, 20.66739718160082551669600870286, 21.52561459792401881746817841261, 22.40018942877772321382133697594, 23.787657121141655755698342436553, 24.870731451950152437115756398902, 25.90940326666516649621457390031, 26.76344750614363758891125194475, 27.28914749016952293952357765310