L(s) = 1 | + (−0.0747 + 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.733 + 0.680i)5-s + (0.222 − 0.974i)8-s + (−0.733 + 0.680i)10-s + (−0.826 − 0.563i)11-s + (−0.900 + 0.433i)13-s + (0.955 + 0.294i)16-s + (−0.365 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.623 − 0.781i)20-s + (0.623 − 0.781i)22-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.365 − 0.930i)26-s + ⋯ |
L(s) = 1 | + (−0.0747 + 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.733 + 0.680i)5-s + (0.222 − 0.974i)8-s + (−0.733 + 0.680i)10-s + (−0.826 − 0.563i)11-s + (−0.900 + 0.433i)13-s + (0.955 + 0.294i)16-s + (−0.365 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.623 − 0.781i)20-s + (0.623 − 0.781i)22-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.365 − 0.930i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1643137304 + 0.2710526849i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1643137304 + 0.2710526849i\) |
\(L(1)\) |
\(\approx\) |
\(0.6068039247 + 0.4430444893i\) |
\(L(1)\) |
\(\approx\) |
\(0.6068039247 + 0.4430444893i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.0747 + 0.997i)T \) |
| 5 | \( 1 + (0.733 + 0.680i)T \) |
| 11 | \( 1 + (-0.826 - 0.563i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| 17 | \( 1 + (-0.365 + 0.930i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.365 - 0.930i)T \) |
| 29 | \( 1 + (-0.623 - 0.781i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.988 + 0.149i)T \) |
| 41 | \( 1 + (0.222 - 0.974i)T \) |
| 43 | \( 1 + (-0.222 - 0.974i)T \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T \) |
| 53 | \( 1 + (0.988 + 0.149i)T \) |
| 59 | \( 1 + (0.733 - 0.680i)T \) |
| 61 | \( 1 + (-0.988 + 0.149i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.0747 + 0.997i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.900 + 0.433i)T \) |
| 89 | \( 1 + (-0.826 + 0.563i)T \) |
| 97 | \( 1 + T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.61284599227213653821803989651, −26.41162852288026438637503305234, −25.46027591841176124867487126597, −24.24495651297950947763922705279, −23.166211211857080607966286219576, −22.01423816303001961011558780535, −21.279265159510193639084823892216, −20.28326627625354249438060331326, −19.62295205271742177278884565481, −18.06022056277613772119369138329, −17.66131233303724574432778533819, −16.390959347397110438979268322643, −14.89875059210527499457466819669, −13.57030954548728418697833407007, −12.907413007966607373124511503892, −11.91713341793497662686946739314, −10.57120338817236836578943444378, −9.6715040454263509523516642830, −8.78051831209663550339359799307, −7.36926902248786952887986563385, −5.36994237158208266651574541012, −4.68633168642923456822767937652, −2.88188704191740949481711125784, −1.77589761709716038689037095112, −0.11662559917347098874539480307,
2.19029817327759210482988907906, 3.95331182656333224826774926372, 5.46382620977225317274767974, 6.31183742725239979979479398670, 7.45940656508831650013242039990, 8.60308969788404365948163894995, 9.87447814901301528068362608500, 10.704924418756836521801886353488, 12.56063553980061487057449939821, 13.62158589278141853270237516663, 14.51048244029233651112807545730, 15.35058952680080459812333806295, 16.67582378497440442815551611435, 17.38066532326463018882517804150, 18.54705880257312675778567396194, 19.12617395700853083990550918626, 20.95937931499895623425567362754, 21.94729979344030118107611072348, 22.68478660105913656622860418427, 23.94585896613118477823334937128, 24.606308804097184801723003553835, 25.87977226776734560068002044032, 26.312911185268235140474275542564, 27.27365395195596133529121638260, 28.57200254951345092668846642954