L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.173 − 0.984i)6-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.766 − 0.642i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 + 0.984i)13-s + (−0.766 + 0.642i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (0.5 − 0.866i)18-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.173 − 0.984i)6-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.766 − 0.642i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 + 0.984i)13-s + (−0.766 + 0.642i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (0.5 − 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7304388418 + 0.4570063137i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7304388418 + 0.4570063137i\) |
\(L(1)\) |
\(\approx\) |
\(0.8484550623 + 0.2061516794i\) |
\(L(1)\) |
\(\approx\) |
\(0.8484550623 + 0.2061516794i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.173 + 0.984i)T \) |
| 17 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.939 + 0.342i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.766 + 0.642i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (-0.173 - 0.984i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.766 + 0.642i)T \) |
| 71 | \( 1 + (-0.766 - 0.642i)T \) |
| 73 | \( 1 + (-0.766 - 0.642i)T \) |
| 79 | \( 1 + (0.939 - 0.342i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.939 + 0.342i)T \) |
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
−28.31901571272632617144601996926, −27.369277376266445310614330113673, −26.3166233204475923956896143731, −25.22375991737112725485560893547, −24.84655761552649559185750041999, −23.75760931298049017000878439261, −22.976099254828700084165140124480, −20.814959325639346293840421999495, −20.17598800875553282106753680028, −19.38523935157218485689341455714, −18.06061479880867284208135542229, −17.551133611855775449122025982, −15.999454438211881764562874550712, −15.36871336724365201160945361420, −14.05922260717502990746304508585, −13.02842773282119756901545685311, −11.8882151580764025235839138486, −10.090778068062167565233585700604, −9.15661920826201547126157316815, −8.07392483427464086439557925224, −7.459087624461769129777871031170, −5.96656363997272870387041751419, −4.58483813553486446330083380727, −2.46141894578603389004761104653, −0.96030141824756880495517419065,
2.123861997389562748820656866246, 3.20220396557323551966891742074, 4.21470781046753789125870675003, 6.48162592793954999743313600485, 7.90928601116273103686078479562, 8.68825327510458225809583154832, 10.02456981422393989865215188086, 10.69958092226543601939957936395, 11.73520788739013841752780099652, 13.381521611709030378458425784670, 14.34153668139799832430298215837, 15.65106640106129314481369256547, 16.46704936790884811078547089700, 17.891049316654075122155086516119, 19.05301602138547783605158245178, 19.41243011299455079937398849343, 20.76846039875575863703100706622, 21.601267021657753677935920148549, 22.24636048281969943174030157969, 23.89378793599986626484999323524, 25.34606529159045553721380313869, 26.41416775488168987292531323621, 26.47736002717050469993060288255, 27.64902763114151625836676646842, 28.68695002517213141185869127918