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
−22.59523314487437483783255660336, −22.02976787873965497688282405630, −21.19996959739233779635917132630, −20.74062061051040237720263716307, −19.06200888852964999355076886446, −18.479789260249172734072304304626, −17.98440590490159162766836116697, −17.0416784143725054307739712114, −16.27495815970161750478520878526, −15.28383310697231821408934480844, −14.76771531723921471554026212194, −13.435835499359115631463014266938, −12.82085643972573847571261851928, −11.715241143037276353959576843466, −11.10909656094886386487806583816, −10.14429760479777096134074292923, −9.60008629890204139769184453782, −8.3400698011090740833349548996, −7.263526882377527080808573326346, −6.14838914590635592943938741386, −5.608884659566550089127697676360, −4.95765876893135606396809251819, −3.31491093620299091914876963517, −2.51508972004701331059500619566, −1.12274955280912195352249659027,
0.27936719428436501010005271741, 1.356960127494709218149796738312, 2.24810628171882024068192896019, 4.1303541033978073857359200220, 4.7660591291509603885740918574, 5.620520361908428638309005809923, 6.72683161088883905848850674650, 7.27179463081335881747552862946, 8.53400378369838406167210977002, 9.67311659646014602780613917559, 10.35822890016640122005769332869, 11.109715016354358769146400055565, 12.18704087116420871555968762732, 13.021708072681027222260021480653, 13.44252361077047037432062102944, 14.60744649816715005991089387678, 15.69354902466706877350155566772, 16.7579977819492083855374696695, 17.061151188735001005704164940715, 17.72299884390004593307392542465, 18.6854962447461752115467733147, 19.619863889971441219131436142021, 20.63016657324322250420688523549, 21.30736701533366070728225587040, 21.98250502107264949926502302021