L(s) = 1 | + (0.540 − 0.841i)2-s + (0.989 − 0.142i)3-s + (−0.415 − 0.909i)4-s + (0.415 − 0.909i)6-s + (0.755 + 0.654i)7-s + (−0.989 − 0.142i)8-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.540 − 0.841i)12-s + (0.755 − 0.654i)13-s + (0.959 − 0.281i)14-s + (−0.654 + 0.755i)16-s + (−0.909 − 0.415i)17-s + (0.281 − 0.959i)18-s + (−0.415 − 0.909i)19-s + ⋯ |
L(s) = 1 | + (0.540 − 0.841i)2-s + (0.989 − 0.142i)3-s + (−0.415 − 0.909i)4-s + (0.415 − 0.909i)6-s + (0.755 + 0.654i)7-s + (−0.989 − 0.142i)8-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.540 − 0.841i)12-s + (0.755 − 0.654i)13-s + (0.959 − 0.281i)14-s + (−0.654 + 0.755i)16-s + (−0.909 − 0.415i)17-s + (0.281 − 0.959i)18-s + (−0.415 − 0.909i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.258448834 - 2.501675482i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.258448834 - 2.501675482i\) |
\(L(1)\) |
\(\approx\) |
\(1.706784585 - 1.106382069i\) |
\(L(1)\) |
\(\approx\) |
\(1.706784585 - 1.106382069i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.540 - 0.841i)T \) |
| 3 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (0.755 + 0.654i)T \) |
| 11 | \( 1 + (0.841 - 0.540i)T \) |
| 13 | \( 1 + (0.755 - 0.654i)T \) |
| 17 | \( 1 + (-0.909 - 0.415i)T \) |
| 19 | \( 1 + (-0.415 - 0.909i)T \) |
| 29 | \( 1 + (-0.415 + 0.909i)T \) |
| 31 | \( 1 + (-0.142 + 0.989i)T \) |
| 37 | \( 1 + (0.281 + 0.959i)T \) |
| 41 | \( 1 + (-0.959 - 0.281i)T \) |
| 43 | \( 1 + (0.989 - 0.142i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.755 - 0.654i)T \) |
| 59 | \( 1 + (0.654 + 0.755i)T \) |
| 61 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (0.540 - 0.841i)T \) |
| 71 | \( 1 + (0.841 + 0.540i)T \) |
| 73 | \( 1 + (-0.909 + 0.415i)T \) |
| 79 | \( 1 + (0.654 + 0.755i)T \) |
| 83 | \( 1 + (-0.281 - 0.959i)T \) |
| 89 | \( 1 + (0.142 + 0.989i)T \) |
| 97 | \( 1 + (-0.281 + 0.959i)T \) |
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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
−29.856943837972436471437846286556, −27.89598010043959300792865971301, −26.89592277305982925530152082113, −26.16663703513649905867853478612, −25.17003221056496404608410155361, −24.34693081485947129148722783686, −23.40938028089395636754388381207, −22.178029963269992733161970048370, −21.04777187340088616911180117441, −20.36306310122952438566925703539, −18.94494578282366141116815979467, −17.640439783628790397297666750782, −16.63530665850209375961870971040, −15.403384278393596948102980928606, −14.51193984735900281478243068384, −13.81833473804989171511705919709, −12.7443548190745328725921990894, −11.224253318793056526956868473983, −9.50730280320474121617464607300, −8.43608767206406195201438520803, −7.48874566391019047004414337893, −6.30959990665502515347993373736, −4.39229578203064650725705493152, −3.87785926249197337156790678445, −1.90256197441717797083827263132,
1.267591019252301479573348953645, 2.545244637160982823960581689724, 3.731679883384624671579740063955, 5.059223490469421991626715895488, 6.62789423653162934340663907638, 8.53563558210233193949576079030, 9.100579308679853846183263590381, 10.6861636060174824921709333897, 11.71601682185934532096795815052, 12.968419674179207605845888766444, 13.86152069308602193735790477492, 14.82344490006350713569960525848, 15.66692107944939892642537308788, 17.80586508267680431326269614437, 18.647378523659189372172669357747, 19.725032126531049281409046739119, 20.47749000762355749012119914452, 21.498448886523960973342989750807, 22.25289773517225334968078603248, 23.81717947514352432126289548877, 24.54321462414681192934443735310, 25.553226549084925931142764280454, 27.09057658947661733526098881347, 27.66535997662587243179756537548, 28.94017276481996221714540724401