L(s) = 1 | + (−0.817 − 0.576i)2-s + (−0.619 − 0.785i)3-s + (0.336 + 0.941i)4-s + (0.197 + 0.980i)5-s + (0.0541 + 0.998i)6-s + (−0.999 + 0.0361i)7-s + (0.267 − 0.963i)8-s + (−0.232 + 0.972i)9-s + (0.403 − 0.915i)10-s + (−0.161 − 0.986i)11-s + (0.530 − 0.847i)12-s + (−0.856 − 0.515i)13-s + (0.837 + 0.546i)14-s + (0.647 − 0.762i)15-s + (−0.773 + 0.633i)16-s + (0.976 − 0.214i)17-s + ⋯ |
L(s) = 1 | + (−0.817 − 0.576i)2-s + (−0.619 − 0.785i)3-s + (0.336 + 0.941i)4-s + (0.197 + 0.980i)5-s + (0.0541 + 0.998i)6-s + (−0.999 + 0.0361i)7-s + (0.267 − 0.963i)8-s + (−0.232 + 0.972i)9-s + (0.403 − 0.915i)10-s + (−0.161 − 0.986i)11-s + (0.530 − 0.847i)12-s + (−0.856 − 0.515i)13-s + (0.837 + 0.546i)14-s + (0.647 − 0.762i)15-s + (−0.773 + 0.633i)16-s + (0.976 − 0.214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3065542903 - 0.1121609393i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3065542903 - 0.1121609393i\) |
\(L(1)\) |
\(\approx\) |
\(0.4097896217 - 0.1363996817i\) |
\(L(1)\) |
\(\approx\) |
\(0.4097896217 - 0.1363996817i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4003 | \( 1 \) |
good | 2 | \( 1 + (-0.817 - 0.576i)T \) |
| 3 | \( 1 + (-0.619 - 0.785i)T \) |
| 5 | \( 1 + (0.197 + 0.980i)T \) |
| 7 | \( 1 + (-0.999 + 0.0361i)T \) |
| 11 | \( 1 + (-0.161 - 0.986i)T \) |
| 13 | \( 1 + (-0.856 - 0.515i)T \) |
| 17 | \( 1 + (0.976 - 0.214i)T \) |
| 19 | \( 1 + (-0.674 - 0.738i)T \) |
| 23 | \( 1 + (-0.773 + 0.633i)T \) |
| 29 | \( 1 + (-0.674 + 0.738i)T \) |
| 31 | \( 1 + (-0.983 - 0.179i)T \) |
| 37 | \( 1 + (-0.674 + 0.738i)T \) |
| 41 | \( 1 + (-0.161 + 0.986i)T \) |
| 43 | \( 1 + (0.989 + 0.143i)T \) |
| 47 | \( 1 + (-0.773 - 0.633i)T \) |
| 53 | \( 1 + (-0.773 - 0.633i)T \) |
| 59 | \( 1 + (0.700 + 0.713i)T \) |
| 61 | \( 1 + (-0.891 + 0.452i)T \) |
| 67 | \( 1 + (0.837 - 0.546i)T \) |
| 71 | \( 1 + (-0.619 - 0.785i)T \) |
| 73 | \( 1 + (0.796 - 0.605i)T \) |
| 79 | \( 1 + (-0.891 - 0.452i)T \) |
| 83 | \( 1 + (-0.817 - 0.576i)T \) |
| 89 | \( 1 + (0.837 + 0.546i)T \) |
| 97 | \( 1 + (0.837 - 0.546i)T \) |
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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
−18.505736258290753159958670066209, −17.449977406023396498231261322377, −17.13128588948080315703132567279, −16.610063583801076216142180773834, −15.97425830680319681592295018164, −15.5715206774262941333403601847, −14.56629339857786556446330865097, −14.197345575264248391246495631857, −12.677674016283202242986105937754, −12.49870641069014425364624297140, −11.68469165772821820686206927171, −10.56850611703857967711461922453, −10.07255260978079196479678597989, −9.51980786629840575437868474213, −9.14192388414073199305822803790, −8.166758226372402234462811327433, −7.35111440330636852630321504647, −6.56923745736372789193887483408, −5.77020265907218580239183492819, −5.36360724103652001464312778190, −4.412657686511457795058395152780, −3.83540152770583130763197204183, −2.38596119140423098945364763581, −1.58875193914771319424674654224, −0.33619791760661306506237029120,
0.37198674973159351749439053835, 1.548074779244328464441740619768, 2.42654808868521736850122381338, 3.083613621302669652631871546472, 3.61945923711185276271716132153, 5.119133424441198330402981272717, 6.00417613891682954182227946839, 6.52379555046216125699965977328, 7.37918146926601294915573892591, 7.68742161849975849088544418436, 8.6992200772362786767638471921, 9.61294420059742090141363544715, 10.217232524384688772777404954295, 10.79578833016764870924686566236, 11.4613637242805603241965761627, 12.08227055546512075239232682205, 12.886689628573155268215893093686, 13.305841049922428415035883740790, 14.13756753944858201273641343021, 15.09604219344713124931269448288, 16.03309314415151015276485788070, 16.58411142288410773041159285214, 17.18804258795333613604076473477, 17.91555978090638959586610252024, 18.4882893220405539322341219188