L(s) = 1 | + (−0.977 + 0.212i)3-s + (0.755 + 0.654i)5-s + (0.0713 + 0.997i)7-s + (0.909 − 0.415i)9-s + (−0.654 − 0.755i)11-s + (−0.212 − 0.977i)13-s + (−0.877 − 0.479i)15-s + (0.281 − 0.959i)17-s + (−0.349 + 0.936i)19-s + (−0.281 − 0.959i)21-s + (0.936 + 0.349i)23-s + (0.142 + 0.989i)25-s + (−0.800 + 0.599i)27-s + (−0.997 + 0.0713i)29-s + (0.936 − 0.349i)31-s + ⋯ |
L(s) = 1 | + (−0.977 + 0.212i)3-s + (0.755 + 0.654i)5-s + (0.0713 + 0.997i)7-s + (0.909 − 0.415i)9-s + (−0.654 − 0.755i)11-s + (−0.212 − 0.977i)13-s + (−0.877 − 0.479i)15-s + (0.281 − 0.959i)17-s + (−0.349 + 0.936i)19-s + (−0.281 − 0.959i)21-s + (0.936 + 0.349i)23-s + (0.142 + 0.989i)25-s + (−0.800 + 0.599i)27-s + (−0.997 + 0.0713i)29-s + (0.936 − 0.349i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2744195287 + 0.9252587312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2744195287 + 0.9252587312i\) |
\(L(1)\) |
\(\approx\) |
\(0.7910721749 + 0.2519689218i\) |
\(L(1)\) |
\(\approx\) |
\(0.7910721749 + 0.2519689218i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (0.755 + 0.654i)T \) |
| 7 | \( 1 + (0.0713 + 0.997i)T \) |
| 11 | \( 1 + (-0.654 - 0.755i)T \) |
| 13 | \( 1 + (-0.212 - 0.977i)T \) |
| 17 | \( 1 + (0.281 - 0.959i)T \) |
| 19 | \( 1 + (-0.349 + 0.936i)T \) |
| 23 | \( 1 + (0.936 + 0.349i)T \) |
| 29 | \( 1 + (-0.997 + 0.0713i)T \) |
| 31 | \( 1 + (0.936 - 0.349i)T \) |
| 37 | \( 1 + (0.707 + 0.707i)T \) |
| 41 | \( 1 + (-0.212 + 0.977i)T \) |
| 43 | \( 1 + (-0.997 - 0.0713i)T \) |
| 47 | \( 1 + (0.540 + 0.841i)T \) |
| 53 | \( 1 + (0.540 - 0.841i)T \) |
| 59 | \( 1 + (0.977 + 0.212i)T \) |
| 61 | \( 1 + (0.599 + 0.800i)T \) |
| 67 | \( 1 + (-0.841 - 0.540i)T \) |
| 71 | \( 1 + (0.755 - 0.654i)T \) |
| 73 | \( 1 + (-0.415 + 0.909i)T \) |
| 79 | \( 1 + (-0.909 - 0.415i)T \) |
| 83 | \( 1 + (-0.877 + 0.479i)T \) |
| 97 | \( 1 + (-0.654 + 0.755i)T \) |
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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
−21.98250502107264949926502302021, −21.30736701533366070728225587040, −20.63016657324322250420688523549, −19.619863889971441219131436142021, −18.6854962447461752115467733147, −17.72299884390004593307392542465, −17.061151188735001005704164940715, −16.7579977819492083855374696695, −15.69354902466706877350155566772, −14.60744649816715005991089387678, −13.44252361077047037432062102944, −13.021708072681027222260021480653, −12.18704087116420871555968762732, −11.109715016354358769146400055565, −10.35822890016640122005769332869, −9.67311659646014602780613917559, −8.53400378369838406167210977002, −7.27179463081335881747552862946, −6.72683161088883905848850674650, −5.620520361908428638309005809923, −4.7660591291509603885740918574, −4.1303541033978073857359200220, −2.24810628171882024068192896019, −1.356960127494709218149796738312, −0.27936719428436501010005271741,
1.12274955280912195352249659027, 2.51508972004701331059500619566, 3.31491093620299091914876963517, 4.95765876893135606396809251819, 5.608884659566550089127697676360, 6.14838914590635592943938741386, 7.263526882377527080808573326346, 8.3400698011090740833349548996, 9.60008629890204139769184453782, 10.14429760479777096134074292923, 11.10909656094886386487806583816, 11.715241143037276353959576843466, 12.82085643972573847571261851928, 13.435835499359115631463014266938, 14.76771531723921471554026212194, 15.28383310697231821408934480844, 16.27495815970161750478520878526, 17.0416784143725054307739712114, 17.98440590490159162766836116697, 18.479789260249172734072304304626, 19.06200888852964999355076886446, 20.74062061051040237720263716307, 21.19996959739233779635917132630, 22.02976787873965497688282405630, 22.59523314487437483783255660336