L(s) = 1 | + (0.580 − 0.814i)2-s + (0.723 + 0.690i)3-s + (−0.327 − 0.945i)4-s + (−0.142 − 0.989i)5-s + (0.981 − 0.189i)6-s + (0.235 + 0.971i)7-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (−0.888 − 0.458i)10-s + (0.841 − 0.540i)11-s + (0.415 − 0.909i)12-s + (0.580 − 0.814i)13-s + (0.928 + 0.371i)14-s + (0.580 − 0.814i)15-s + (−0.786 + 0.618i)16-s + (0.841 − 0.540i)17-s + ⋯ |
L(s) = 1 | + (0.580 − 0.814i)2-s + (0.723 + 0.690i)3-s + (−0.327 − 0.945i)4-s + (−0.142 − 0.989i)5-s + (0.981 − 0.189i)6-s + (0.235 + 0.971i)7-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (−0.888 − 0.458i)10-s + (0.841 − 0.540i)11-s + (0.415 − 0.909i)12-s + (0.580 − 0.814i)13-s + (0.928 + 0.371i)14-s + (0.580 − 0.814i)15-s + (−0.786 + 0.618i)16-s + (0.841 − 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.614029207 - 1.001965757i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.614029207 - 1.001965757i\) |
\(L(1)\) |
\(\approx\) |
\(1.529182715 - 0.6299908188i\) |
\(L(1)\) |
\(\approx\) |
\(1.529182715 - 0.6299908188i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (0.580 - 0.814i)T \) |
| 3 | \( 1 + (0.723 + 0.690i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.235 + 0.971i)T \) |
| 11 | \( 1 + (0.841 - 0.540i)T \) |
| 13 | \( 1 + (0.580 - 0.814i)T \) |
| 17 | \( 1 + (0.841 - 0.540i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.327 + 0.945i)T \) |
| 29 | \( 1 + (-0.995 + 0.0950i)T \) |
| 31 | \( 1 + (0.723 - 0.690i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.0475 + 0.998i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.235 + 0.971i)T \) |
| 53 | \( 1 + (0.580 + 0.814i)T \) |
| 59 | \( 1 + (-0.654 - 0.755i)T \) |
| 61 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (-0.142 - 0.989i)T \) |
| 71 | \( 1 + (0.928 - 0.371i)T \) |
| 73 | \( 1 + (-0.327 + 0.945i)T \) |
| 79 | \( 1 + (-0.786 + 0.618i)T \) |
| 83 | \( 1 + (0.415 + 0.909i)T \) |
| 89 | \( 1 + (-0.995 + 0.0950i)T \) |
| 97 | \( 1 + (0.723 - 0.690i)T \) |
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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
−26.500605750821779887024817245024, −26.12430848056794098399453781117, −25.21291075767289962288554688190, −24.25725864884082699017366321903, −23.25267615631887749183227690954, −22.86848084443826582857648630774, −21.42716117699778402068736568762, −20.5525907826960708070679529069, −19.324342950545717227587897818710, −18.42842541888799020486965015351, −17.4290095640502303233221973685, −16.473214594357433337523901778187, −15.01230732517269051507743775273, −14.35535950317727785560430845582, −13.88902855354981581938638648501, −12.63189020081893721939930348019, −11.65695713191554668788359076359, −10.15561922082063862686791625167, −8.707904665596631337187903446255, −7.67793447681042502097812185706, −6.894396279570198827443223804954, −6.17638151791800804766412622156, −4.09316338295692213730741363224, −3.542787304519802509121973582911, −1.87838138330266175962477605308,
1.42087280471828429740674626620, 2.8543698537646859674783604727, 3.88139974133997510471207276284, 5.01769866361624258244869134344, 5.83341753320347494738908605061, 8.13480703007469284882725829367, 9.02214871316891839060724104059, 9.69815559843907468123828877338, 11.1374659987652356068705855900, 11.9700101113402708710393932991, 13.125155702113022465074333568192, 13.92604170144206744739388322711, 15.14347437405578963598595039386, 15.68965324977363483746442074039, 17.02202633787822343888157443158, 18.544669331101533860429371556526, 19.46782072513394509945619987875, 20.26761361155386905880172150112, 21.06056854675298611142306661703, 21.740164597710618688874208959703, 22.64566274354716819423935728585, 23.98945554830475491448938179572, 24.80477136074031718681919312076, 25.60577807042033109419935141014, 27.290500295420753264550985183016