L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)10-s + (0.766 − 0.642i)11-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s − 26-s + ⋯ |
L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)10-s + (0.766 − 0.642i)11-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s − 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9449824331 + 1.061868170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9449824331 + 1.061868170i\) |
\(L(1)\) |
\(\approx\) |
\(0.8721767601 + 0.4544349927i\) |
\(L(1)\) |
\(\approx\) |
\(0.8721767601 + 0.4544349927i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 11 | \( 1 + (0.766 - 0.642i)T \) |
| 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (0.939 - 0.342i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.939 + 0.342i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (-0.766 - 0.642i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.173 - 0.984i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.766 + 0.642i)T \) |
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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
−26.82036209318733011174304647115, −26.006920862499184405498906501878, −24.56211711986583116105918132713, −23.43343407977958346653871149459, −22.62094611099606153653761368386, −22.041371078352783722441670867124, −20.773057999879623892004646291376, −19.7535279687335170188326421032, −19.30644340732628783421731393018, −18.06390608019670029350108852106, −17.328738094982450260073761549968, −15.6015337617244265799735301667, −14.77406600366141582469459404779, −13.81202982158277897826462307282, −12.44913974880553457086891781382, −11.85885694025496969581051597760, −10.69328740888468425805811578972, −9.93295651647402350208836039233, −8.5725623002895346431808349515, −7.43894215795591371586824302354, −5.945539912031720610878746614340, −4.45841678771590888241331342073, −3.52832142697946085923882399755, −2.33528838904625731178008082791, −0.63750291062728615536334056400,
0.99959409974577457520193066130, 3.47827226333233086920371976973, 4.42529937636873165542124942534, 5.57485567280182399007325156792, 6.813151829615224983422830417501, 7.86585486115522842339942488589, 8.79682174405627676483373996951, 9.7438313533541892636524861731, 11.66816638778418994404866124993, 12.2730064409639222833539135518, 13.73206077872291086873418093519, 14.36544100434211776263934824357, 15.67611606531823561186922839805, 16.4003624161119732468472621687, 17.04326823133584449052720354814, 18.45578752248828116418220014783, 19.248493747265450875875215171, 20.47868711277352146823759796312, 21.645083617676435974523325783100, 22.60870938245876283183443529031, 23.56498417590376918887047905733, 24.3117129946777079899830150005, 25.011915015584654036670290970802, 26.17649354667686058995330424308, 27.17706525559917959263305095632