L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s − 18-s + (−0.939 + 0.342i)21-s + (−0.173 + 0.984i)22-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s − 18-s + (−0.939 + 0.342i)21-s + (−0.173 + 0.984i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.571692994 + 0.2899569907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.571692994 + 0.2899569907i\) |
\(L(1)\) |
\(\approx\) |
\(1.578159458 + 0.1664993896i\) |
\(L(1)\) |
\(\approx\) |
\(1.578159458 + 0.1664993896i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.173 - 0.984i)T \) |
| 17 | \( 1 + (0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.939 + 0.342i)T \) |
| 53 | \( 1 + (-0.766 + 0.642i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.173 + 0.984i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.939 - 0.342i)T \) |
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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
−30.098273250519332063081516386406, −29.57925368913104635764749910676, −28.44699283325503850813464615500, −26.66076109625168621466219535293, −25.77432886185773253347660027443, −24.45993883390187835403735074497, −23.85767516458756251124195992460, −23.175044943641470856167296800851, −21.84284817638075760753100048494, −20.765224934983019215067838981795, −19.618010660093505292210271669927, −18.36651169793960011929371083784, −16.99737708059424991065365713823, −16.34021950856856501891259533399, −14.4736576849830076244645743652, −13.9054061034512150883355944988, −12.82961738240976584427396797173, −11.71619309511336784301089594127, −10.72958036012125169169724160348, −8.32858434780013019276724817197, −7.39785994433572317732510582948, −6.30175843895685207422135691777, −5.05282497212988865566950424601, −3.46136663112393954914399035571, −1.76632440623587475050293720066,
2.30929494499540452280877220539, 3.67639323412865879741081794284, 5.11564430202304506665285428883, 5.698043794023226827710570017105, 7.714465745071805576471901776856, 9.52017565619962028900128537477, 10.497172697587405053648741958646, 11.68532749793297873508842668555, 12.58695125626857444353258068218, 14.17790823292114283677645583338, 15.20978044189647898871208352133, 15.71295794635145761170139075103, 17.26393434299206033266490828115, 18.65755477301279328288338449285, 20.26840475087088801799861468145, 20.81764171262074136620427107979, 21.90724193126677750095452628855, 22.63744711921003035556577046608, 23.68009492002541015124627109655, 25.01366945189344378631744064477, 25.89989367323361328665123223969, 27.72142283242071668724777519826, 27.984666376661910173003331734941, 29.24936978158029641501085661698, 30.42335578433688237400055247932