L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.939 + 0.342i)3-s + (−0.939 + 0.342i)4-s + (−0.342 + 0.939i)5-s + (0.173 − 0.984i)6-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s − 12-s + (−0.766 + 0.642i)13-s + (−0.173 + 0.984i)14-s + (−0.642 + 0.766i)15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.939 + 0.342i)3-s + (−0.939 + 0.342i)4-s + (−0.342 + 0.939i)5-s + (0.173 − 0.984i)6-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s − 12-s + (−0.766 + 0.642i)13-s + (−0.173 + 0.984i)14-s + (−0.642 + 0.766i)15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9608418728 - 0.5050341441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9608418728 - 0.5050341441i\) |
\(L(1)\) |
\(\approx\) |
\(0.8762209677 - 0.1807674340i\) |
\(L(1)\) |
\(\approx\) |
\(0.8762209677 - 0.1807674340i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.342 + 0.939i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (-0.984 + 0.173i)T \) |
| 13 | \( 1 + (-0.766 + 0.642i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (0.939 - 0.342i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.642 - 0.766i)T \) |
| 31 | \( 1 + (-0.642 - 0.766i)T \) |
| 41 | \( 1 + (-0.984 - 0.173i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + (0.342 - 0.939i)T \) |
| 53 | \( 1 + (0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.766 - 0.642i)T \) |
| 61 | \( 1 + (-0.342 + 0.939i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (0.939 + 0.342i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (0.342 + 0.939i)T \) |
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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
−18.38303276616161270247350243907, −18.07062565568325885217081864729, −17.07079282592045548739911936047, −16.37523229043329439967164573600, −15.73147678601700650489723923715, −15.35012201105444575082559426210, −14.71273044982772036291227726034, −13.60813022590611402186922010658, −13.33579034341308929980242570337, −12.55621581641436672665115282479, −12.284314610969999381477850835985, −10.71765261708795907502909160391, −9.92016003251987137925762454856, −9.28496957980611495289108707035, −8.72732582350155461490972692955, −8.15476656489181115773513582387, −7.42853304854709855164421326177, −6.89570594313920990669038815690, −5.92493622102669139195410379390, −5.1213149388722503538371688139, −4.57161235893240027592768576198, −3.40813857851323794332991475108, −2.950195401404566499863806006279, −1.668552396685876470449161794353, −0.66826300897500836741151600673,
0.42396114414947642337148227998, 1.995415119002647374491088659152, 2.57730380154548147768927921428, 3.09760202441486162273034076503, 3.78920727739204309460325544422, 4.54116033223698469802977718180, 5.297110078193523259794648711117, 6.795098407424409173082041335659, 7.260076507428472921165478158791, 7.954427938919740003924371754747, 8.90647957876260395929697794645, 9.50801230120753305942213208475, 10.12110646194822204628709105146, 10.570222590500921237834086702577, 11.40457615881894174315918192304, 12.10017796241063823067558544878, 13.10920715065603566138047513250, 13.52010129853329157061112401664, 14.015013438740083704356823551855, 15.04010924239680274654420166318, 15.416699907954118832389570848259, 16.29801121374960389517718024360, 17.05001187805144813749379336555, 18.07957115552918433333906613305, 18.61485633007128436968222823902