L(s) = 1 | + (−0.0348 + 0.999i)2-s + (0.719 + 0.694i)3-s + (−0.997 − 0.0697i)4-s + (−0.719 + 0.694i)6-s + (−0.5 + 0.866i)7-s + (0.104 − 0.994i)8-s + (0.0348 + 0.999i)9-s + (0.669 − 0.743i)11-s + (−0.669 − 0.743i)12-s + (0.882 + 0.469i)13-s + (−0.848 − 0.529i)14-s + (0.990 + 0.139i)16-s + (0.559 + 0.829i)17-s − 18-s + (−0.961 + 0.275i)21-s + (0.719 + 0.694i)22-s + ⋯ |
L(s) = 1 | + (−0.0348 + 0.999i)2-s + (0.719 + 0.694i)3-s + (−0.997 − 0.0697i)4-s + (−0.719 + 0.694i)6-s + (−0.5 + 0.866i)7-s + (0.104 − 0.994i)8-s + (0.0348 + 0.999i)9-s + (0.669 − 0.743i)11-s + (−0.669 − 0.743i)12-s + (0.882 + 0.469i)13-s + (−0.848 − 0.529i)14-s + (0.990 + 0.139i)16-s + (0.559 + 0.829i)17-s − 18-s + (−0.961 + 0.275i)21-s + (0.719 + 0.694i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.6748890716 + 1.610507208i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.6748890716 + 1.610507208i\) |
\(L(1)\) |
\(\approx\) |
\(0.6497581168 + 0.9519090903i\) |
\(L(1)\) |
\(\approx\) |
\(0.6497581168 + 0.9519090903i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.0348 + 0.999i)T \) |
| 3 | \( 1 + (0.719 + 0.694i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.669 - 0.743i)T \) |
| 13 | \( 1 + (0.882 + 0.469i)T \) |
| 17 | \( 1 + (0.559 + 0.829i)T \) |
| 23 | \( 1 + (-0.374 + 0.927i)T \) |
| 29 | \( 1 + (-0.559 + 0.829i)T \) |
| 31 | \( 1 + (-0.913 + 0.406i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.990 - 0.139i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.559 - 0.829i)T \) |
| 53 | \( 1 + (0.997 + 0.0697i)T \) |
| 59 | \( 1 + (0.615 - 0.788i)T \) |
| 61 | \( 1 + (-0.374 + 0.927i)T \) |
| 67 | \( 1 + (-0.961 - 0.275i)T \) |
| 71 | \( 1 + (0.241 - 0.970i)T \) |
| 73 | \( 1 + (-0.882 + 0.469i)T \) |
| 79 | \( 1 + (0.719 + 0.694i)T \) |
| 83 | \( 1 + (0.913 - 0.406i)T \) |
| 89 | \( 1 + (-0.990 + 0.139i)T \) |
| 97 | \( 1 + (-0.961 + 0.275i)T \) |
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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
−22.944672170974611538111458364939, −22.394469364989883798653419310235, −20.8708613534618626887328901742, −20.4689370658523661673136847457, −19.80178591605449383136859759088, −18.96937181289898320079247459439, −18.198781480358039135858000062885, −17.421339451653004969805624415666, −16.34397444076218596960978728947, −14.90166448284175765703385048606, −14.09796851660024080747911981854, −13.37680741637066030509658697564, −12.65270332940038160130017937242, −11.84016452197817058190576603266, −10.716154664713370472686432830305, −9.73860696351386067560624451549, −9.06394572747842255228080627045, −7.943006983603067015310707263184, −7.11009174609559969422908179901, −5.884330524795705893599832818171, −4.24891994371534018333358667190, −3.57948578776115634814764871372, −2.51879316624099149951696427658, −1.370537627518036207124285487501, −0.43861757176242721093273603712,
1.595185235504564454688319400470, 3.38875926425456870140337812466, 3.84986246311368956352529404591, 5.31975635016984864013506164277, 5.99038042698133419963760361659, 7.13596317448928071286631504901, 8.43587537013917649508907655367, 8.82468558083369874594692122819, 9.63804922409124778919028732673, 10.69738446712426353930577370479, 12.03570278754547247493986138538, 13.25680096895109827776929981425, 13.92933670236382701252100758549, 14.84066674937347968828937321014, 15.48482247462657880816140116839, 16.34054052286937391398274831487, 16.8418271552860202634123362654, 18.24507927580679197398772344832, 18.99708320144725352623258446388, 19.634298547304755939112191623273, 20.97637538338913639704737978705, 21.93661252427173431345518599963, 22.136356768913420334229222885996, 23.49013924128714858427230656980, 24.2393101416362032924115890810