L(s) = 1 | + (0.994 − 0.104i)2-s + (−0.978 − 0.207i)3-s + (0.978 − 0.207i)4-s + (0.587 − 0.809i)5-s + (−0.994 − 0.104i)6-s + (0.207 + 0.978i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.309 + 0.951i)14-s + (−0.743 + 0.669i)15-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (0.951 + 0.309i)18-s + (0.743 + 0.669i)19-s + ⋯ |
L(s) = 1 | + (0.994 − 0.104i)2-s + (−0.978 − 0.207i)3-s + (0.978 − 0.207i)4-s + (0.587 − 0.809i)5-s + (−0.994 − 0.104i)6-s + (0.207 + 0.978i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.309 + 0.951i)14-s + (−0.743 + 0.669i)15-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (0.951 + 0.309i)18-s + (0.743 + 0.669i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.813910013 - 1.091710121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.813910013 - 1.091710121i\) |
\(L(1)\) |
\(\approx\) |
\(1.759511573 - 0.3947830909i\) |
\(L(1)\) |
\(\approx\) |
\(1.759511573 - 0.3947830909i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.994 - 0.104i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
| 7 | \( 1 + (0.207 + 0.978i)T \) |
| 17 | \( 1 + (0.104 - 0.994i)T \) |
| 19 | \( 1 + (0.743 + 0.669i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.669 + 0.743i)T \) |
| 31 | \( 1 + (-0.587 - 0.809i)T \) |
| 37 | \( 1 + (0.743 - 0.669i)T \) |
| 41 | \( 1 + (0.207 - 0.978i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.951 - 0.309i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.207 + 0.978i)T \) |
| 61 | \( 1 + (-0.104 + 0.994i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.994 - 0.104i)T \) |
| 73 | \( 1 + (-0.951 - 0.309i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.406 + 0.913i)T \) |
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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
−28.58162145545574193312295251266, −27.05104518131460714091452745782, −26.19495096474846081158753984339, −25.04376393143963099120312649955, −23.730239857630746218258139040752, −23.30505504111596136600625820185, −22.171712130432159162009988511309, −21.6286134987394452697026968303, −20.58552900373403838969605977079, −19.25377332857024361884299402433, −17.69180239560154663224098076187, −17.10733907340604009956951248494, −15.91254152159014000800191115711, −14.871920668551767341281340232787, −13.7974675109795681750506907898, −12.91319278427998043252768700738, −11.51805744689085287261470427253, −10.80521892641502614535836123120, −9.91053400691672550957345276420, −7.506952315793692336428765229428, −6.6482526539004176562020316145, −5.66420450696413274019439179437, −4.490338276698517743390152211988, −3.2652584618381412145711541997, −1.43515472526922232012110088622,
1.14019727113423447680612225417, 2.49364309515557037523257631501, 4.5011729182471660793539226319, 5.40839235552381417860518150139, 6.07591334873422168780469149536, 7.501105054509121241395031101485, 9.24012721068998988283728078823, 10.57706651187429010634608291733, 11.82497541344986201369084723087, 12.39219893275750038488934274012, 13.3518799576450552892759043586, 14.55064002083596771559869340350, 15.96543479787112893079297146249, 16.503802341876368479743753580273, 17.826778350642400069798818581185, 18.821191422773974879298389989527, 20.37746003032321998841206643451, 21.21170917562736916556226163456, 22.07686129158514184611108214200, 22.85429390981516610940611719964, 24.03621639576631307439046368222, 24.71510010524090179006578409080, 25.36411241220747860864255896530, 27.32486287755151949426994647280, 28.384885560905861965354957443960