L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.365 − 0.930i)3-s + (−0.222 + 0.974i)4-s + (−0.0747 + 0.997i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + (0.900 − 0.433i)8-s + (−0.733 + 0.680i)9-s + (0.826 − 0.563i)10-s + (−0.222 − 0.974i)11-s + (0.988 − 0.149i)12-s + (0.826 + 0.563i)13-s + (0.365 − 0.930i)14-s + (0.955 − 0.294i)15-s + (−0.900 − 0.433i)16-s + (0.0747 + 0.997i)17-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.365 − 0.930i)3-s + (−0.222 + 0.974i)4-s + (−0.0747 + 0.997i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + (0.900 − 0.433i)8-s + (−0.733 + 0.680i)9-s + (0.826 − 0.563i)10-s + (−0.222 − 0.974i)11-s + (0.988 − 0.149i)12-s + (0.826 + 0.563i)13-s + (0.365 − 0.930i)14-s + (0.955 − 0.294i)15-s + (−0.900 − 0.433i)16-s + (0.0747 + 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8651435366 + 0.06948801848i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8651435366 + 0.06948801848i\) |
\(L(1)\) |
\(\approx\) |
\(0.7217042178 - 0.1298897986i\) |
\(L(1)\) |
\(\approx\) |
\(0.7217042178 - 0.1298897986i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 3 | \( 1 + (-0.365 - 0.930i)T \) |
| 5 | \( 1 + (-0.0747 + 0.997i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.826 + 0.563i)T \) |
| 17 | \( 1 + (0.0747 + 0.997i)T \) |
| 19 | \( 1 + (0.733 + 0.680i)T \) |
| 23 | \( 1 + (0.955 + 0.294i)T \) |
| 29 | \( 1 + (-0.365 + 0.930i)T \) |
| 31 | \( 1 + (-0.988 + 0.149i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.623 + 0.781i)T \) |
| 47 | \( 1 + (-0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.826 - 0.563i)T \) |
| 59 | \( 1 + (-0.900 - 0.433i)T \) |
| 61 | \( 1 + (0.988 + 0.149i)T \) |
| 67 | \( 1 + (-0.733 - 0.680i)T \) |
| 71 | \( 1 + (-0.955 + 0.294i)T \) |
| 73 | \( 1 + (-0.826 - 0.563i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.365 + 0.930i)T \) |
| 89 | \( 1 + (-0.365 - 0.930i)T \) |
| 97 | \( 1 + (-0.222 - 0.974i)T \) |
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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
−34.16315745483041110319265840908, −33.04182717058085161933402128704, −32.668757720052749219349546936745, −31.1684169778002132638842081940, −29.07652313595288026284423944071, −28.05099161708062220376992352501, −27.3428211341036078247051242308, −26.21127433799121185342648243674, −24.948018405352451938533337688334, −23.59314478340867033296330902822, −22.783130030088719240112433203244, −20.67169226621048199178024043559, −20.18101400095924006822799283807, −17.98950950929495313539834653172, −17.0662825661934945761851996770, −16.08745144086917331185193158152, −15.058727811266082430131000983121, −13.46383654725152758935925798567, −11.33712876187275470438067024178, −10.012313576869300424364004614477, −8.89056241174787244984262003096, −7.39921954067244541480225982384, −5.406330071032371610959356331455, −4.43150706577305845615998160846, −0.75464555091294477432823711777,
1.63070017337273275805964874948, 3.19215406777822849195728449990, 5.93940046623998932229186052, 7.56884869833070360760553074196, 8.78435852999417849195104907300, 10.8794585472712594171868841746, 11.50051511632705199824514461330, 12.91492664909831618456122632203, 14.32832533174625196141207671720, 16.37158444483301817492120608395, 17.94013459789881565051283653998, 18.587302004619019220488286445245, 19.37827030235949129315061426880, 21.24206345648156518775906895738, 22.20532197976015736770184812606, 23.5797988055065721053765380296, 25.09231047124840736732217712371, 26.21449532916947793130998054748, 27.51802825635334122334605009009, 28.70887704764327005570151571803, 29.599132459400913432967063974995, 30.75019460653250636441506799153, 31.26634314868370729338436227609, 33.7166331981691213843422078427, 34.86256969578300603120038348830