L(s) = 1 | + (−0.907 − 0.419i)2-s + (−0.947 + 0.319i)3-s + (0.647 + 0.762i)4-s + (−0.856 + 0.515i)5-s + (0.994 + 0.108i)6-s + (0.0541 + 0.998i)7-s + (−0.267 − 0.963i)8-s + (0.796 − 0.605i)9-s + (0.994 − 0.108i)10-s + (0.161 + 0.986i)11-s + (−0.856 − 0.515i)12-s + (−0.796 − 0.605i)13-s + (0.370 − 0.928i)14-s + (0.647 − 0.762i)15-s + (−0.161 + 0.986i)16-s + (0.0541 − 0.998i)17-s + ⋯ |
L(s) = 1 | + (−0.907 − 0.419i)2-s + (−0.947 + 0.319i)3-s + (0.647 + 0.762i)4-s + (−0.856 + 0.515i)5-s + (0.994 + 0.108i)6-s + (0.0541 + 0.998i)7-s + (−0.267 − 0.963i)8-s + (0.796 − 0.605i)9-s + (0.994 − 0.108i)10-s + (0.161 + 0.986i)11-s + (−0.856 − 0.515i)12-s + (−0.796 − 0.605i)13-s + (0.370 − 0.928i)14-s + (0.647 − 0.762i)15-s + (−0.161 + 0.986i)16-s + (0.0541 − 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08303106942 - 0.1363355763i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08303106942 - 0.1363355763i\) |
\(L(1)\) |
\(\approx\) |
\(0.3863772968 + 0.01252683903i\) |
\(L(1)\) |
\(\approx\) |
\(0.3863772968 + 0.01252683903i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (-0.907 - 0.419i)T \) |
| 3 | \( 1 + (-0.947 + 0.319i)T \) |
| 5 | \( 1 + (-0.856 + 0.515i)T \) |
| 7 | \( 1 + (0.0541 + 0.998i)T \) |
| 11 | \( 1 + (0.161 + 0.986i)T \) |
| 13 | \( 1 + (-0.796 - 0.605i)T \) |
| 17 | \( 1 + (0.0541 - 0.998i)T \) |
| 19 | \( 1 + (-0.725 - 0.687i)T \) |
| 23 | \( 1 + (-0.976 - 0.214i)T \) |
| 29 | \( 1 + (0.907 - 0.419i)T \) |
| 31 | \( 1 + (0.725 - 0.687i)T \) |
| 37 | \( 1 + (-0.267 + 0.963i)T \) |
| 41 | \( 1 + (0.976 - 0.214i)T \) |
| 43 | \( 1 + (0.161 - 0.986i)T \) |
| 47 | \( 1 + (0.856 + 0.515i)T \) |
| 53 | \( 1 + (-0.994 - 0.108i)T \) |
| 61 | \( 1 + (-0.907 - 0.419i)T \) |
| 67 | \( 1 + (-0.267 - 0.963i)T \) |
| 71 | \( 1 + (-0.856 - 0.515i)T \) |
| 73 | \( 1 + (0.370 - 0.928i)T \) |
| 79 | \( 1 + (-0.947 - 0.319i)T \) |
| 83 | \( 1 + (0.561 + 0.827i)T \) |
| 89 | \( 1 + (-0.907 + 0.419i)T \) |
| 97 | \( 1 + (0.370 + 0.928i)T \) |
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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
−33.12232347270739384970140594246, −32.0003501180154005449835080333, −30.13568864051915771542774783188, −29.27923030161414287236844611629, −28.194317951145875541594185045509, −27.24404679168582674161017754085, −26.48471321307950756929119222795, −24.659638503286899402205685777985, −23.86205655728338004393443168036, −23.212663269220148660204944901610, −21.37847312306133425730296441887, −19.68443239632213639180147399651, −19.11397028607259197197337951672, −17.53464370072614474117147107581, −16.68670618794806221739688607693, −15.99200355750121205237066879899, −14.2501274433540953275287705158, −12.405766600073577350422043586238, −11.23201365025866709644820673868, −10.231967945620713313342579296862, −8.38521557309953656276993336584, −7.322595215030541333212962059083, −6.05793852553743232729706158145, −4.35761372247536671681390682625, −1.24101200139244016074782577701,
0.14602617910164970444116118564, 2.59751564243716520936730598460, 4.53256056693696848064569538597, 6.517814430140836945364344101714, 7.769188157426725205092639169357, 9.45284013918275056371285092219, 10.57598288368840930862769090394, 11.86998869531146610214965335506, 12.31472460896595765369373156408, 15.19579410970777878974243417783, 15.8077503760827270000641390107, 17.33315118254130587670775496304, 18.1819026845602639553379201723, 19.255717004358478252587523431056, 20.55357526806739994425922492171, 21.970358459907088895928298740944, 22.68218887590127786844695290283, 24.26839165491530551217650016143, 25.59624379574595940626622622131, 26.94288969709111128944652875346, 27.72483441636692380663921508155, 28.404550439942444152721182715591, 29.68677269441781083859234514135, 30.65938993617036171081894442346, 31.98269000261680271714325724095