L(s) = 1 | + (−0.831 + 0.555i)5-s + (0.923 + 0.382i)7-s + (−0.980 − 0.195i)11-s + (−0.831 − 0.555i)13-s + (0.707 − 0.707i)17-s + (0.555 − 0.831i)19-s + (0.382 + 0.923i)23-s + (0.382 − 0.923i)25-s + (0.980 − 0.195i)29-s + i·31-s + (−0.980 + 0.195i)35-s + (−0.555 − 0.831i)37-s + (−0.382 − 0.923i)41-s + (−0.195 + 0.980i)43-s + (−0.707 + 0.707i)47-s + ⋯ |
L(s) = 1 | + (−0.831 + 0.555i)5-s + (0.923 + 0.382i)7-s + (−0.980 − 0.195i)11-s + (−0.831 − 0.555i)13-s + (0.707 − 0.707i)17-s + (0.555 − 0.831i)19-s + (0.382 + 0.923i)23-s + (0.382 − 0.923i)25-s + (0.980 − 0.195i)29-s + i·31-s + (−0.980 + 0.195i)35-s + (−0.555 − 0.831i)37-s + (−0.382 − 0.923i)41-s + (−0.195 + 0.980i)43-s + (−0.707 + 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.616175849 + 0.1993362785i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.616175849 + 0.1993362785i\) |
\(L(1)\) |
\(\approx\) |
\(1.007081209 + 0.07829955382i\) |
\(L(1)\) |
\(\approx\) |
\(1.007081209 + 0.07829955382i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.831 + 0.555i)T \) |
| 7 | \( 1 + (0.923 + 0.382i)T \) |
| 11 | \( 1 + (-0.980 - 0.195i)T \) |
| 13 | \( 1 + (-0.831 - 0.555i)T \) |
| 17 | \( 1 + (0.707 - 0.707i)T \) |
| 19 | \( 1 + (0.555 - 0.831i)T \) |
| 23 | \( 1 + (0.382 + 0.923i)T \) |
| 29 | \( 1 + (0.980 - 0.195i)T \) |
| 31 | \( 1 + iT \) |
| 37 | \( 1 + (-0.555 - 0.831i)T \) |
| 41 | \( 1 + (-0.382 - 0.923i)T \) |
| 43 | \( 1 + (-0.195 + 0.980i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.980 + 0.195i)T \) |
| 59 | \( 1 + (0.831 - 0.555i)T \) |
| 61 | \( 1 + (0.195 + 0.980i)T \) |
| 67 | \( 1 + (0.195 + 0.980i)T \) |
| 71 | \( 1 + (0.923 + 0.382i)T \) |
| 73 | \( 1 + (0.923 - 0.382i)T \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 + (0.555 - 0.831i)T \) |
| 89 | \( 1 + (0.382 - 0.923i)T \) |
| 97 | \( 1 + iT \) |
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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
−24.11625517500847452493592739328, −23.57016316199371740127413884620, −22.73075088594308913483234372883, −21.41937806906940832086494847327, −20.755544104832604904347612619, −20.0257335900005584634729717086, −18.99529135092401591517480947049, −18.2196183856438819710855257822, −16.97779089809865510595710142056, −16.50679226372460606339714420259, −15.282297361709680356091509720855, −14.63259391933379625261318278323, −13.54680521840761722792106317308, −12.39649915766377880912342953843, −11.822316356278014853531920267785, −10.70351518157182673656033943412, −9.83002212020972549221757779464, −8.34150223179178352788705618169, −7.94678418198404842412059325642, −6.90360778629210760392188172363, −5.27354146069274675591417810269, −4.64966266249636659331201476750, −3.53899234216175068095752437585, −2.03468795195088243858968970930, −0.72710152732395946722029175300,
0.729361763714265569174132607086, 2.48070509368867572899325213004, 3.28441343474928428586314774438, 4.82341954659545789035769682235, 5.415440216554138967848830154320, 7.10030920515792657006298041478, 7.69982236300385376475402950084, 8.58985674423033362257128680458, 9.93128278941798033587352333425, 10.89751449404745670379350266001, 11.67933683653465055000978727371, 12.47728452100792715788562194946, 13.78337837867837260998297567975, 14.639862801552047063240801505705, 15.48768169487428655438527602864, 16.09917917283696331880377772628, 17.60227276077140389623144561327, 18.08717588070122381267689600571, 19.11285942090855803293315555133, 19.85816853572216182724465079408, 20.94440409026426838992874919977, 21.66234326735917831487015719910, 22.70301743417108861242276134303, 23.47175245827413748290675524930, 24.24329266852034720858947772780