L(s) = 1 | + (−0.766 − 0.642i)3-s + (−0.5 − 0.866i)7-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.766 − 0.642i)13-s + (−0.173 + 0.984i)17-s + (−0.173 + 0.984i)21-s + (−0.939 − 0.342i)23-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s + (0.939 − 0.342i)33-s + 37-s − 39-s + (0.766 + 0.642i)41-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)3-s + (−0.5 − 0.866i)7-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.766 − 0.642i)13-s + (−0.173 + 0.984i)17-s + (−0.173 + 0.984i)21-s + (−0.939 − 0.342i)23-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s + (0.939 − 0.342i)33-s + 37-s − 39-s + (0.766 + 0.642i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4483597821 - 0.7903507931i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4483597821 - 0.7903507931i\) |
\(L(1)\) |
\(\approx\) |
\(0.7210810264 - 0.2115959292i\) |
\(L(1)\) |
\(\approx\) |
\(0.7210810264 - 0.2115959292i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.766 - 0.642i)T \) |
| 17 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.173 - 0.984i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (-0.939 - 0.342i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.766 - 0.642i)T \) |
| 79 | \( 1 + (-0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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
−22.3153460297333289466984117610, −21.78112905857109052219116102602, −21.069678293620060432393284261174, −20.267635842209318407497268680261, −19.010047127698489180138147361208, −18.424088146520876772482431301557, −17.71458986270573391435306076104, −16.46453502514672794967624243281, −16.08679563507125761809940035201, −15.48599387827755485784333510956, −14.36977149292476651519265413129, −13.411998472662316865953850018812, −12.4665012981779713242340667794, −11.542657178031760667985217726340, −11.0869937710253221888513855219, −9.947912124131493913719049447697, −9.22843273059753140861093557850, −8.44361155794849776896971418199, −7.082664358442009498737343668420, −5.99317652062991817837335627269, −5.638749601656674386222859418087, −4.45305744273411885526903080650, −3.48645856649650619405383526501, −2.4527715058657415568643062708, −0.8374469805871667978804411704,
0.3129038487283388362338395071, 1.34772778422372352289145194422, 2.500276816423421237904812811019, 3.88614540952396741168312324143, 4.74959575905133835234462784460, 6.00021442554967283646530852485, 6.48178203876427430121140308338, 7.63640214619462253302535159019, 8.10173574972657820662181781279, 9.65811648438673519660092032117, 10.48299937330604634200476959008, 11.01369266350031060148059313887, 12.22347993954060500909829045811, 12.892681949337525112527599449004, 13.44156347465827999057076964708, 14.475934255025410441886211185242, 15.70902247976167413899073710257, 16.23409151565913960123936844454, 17.3981630119187592424314603112, 17.652995498988567959039949012697, 18.64958385871306177570996808278, 19.498270558757494171163202721090, 20.225883186267290232741303648175, 21.10751091862960329059728473603, 22.24605098875268417926506122758