L(s) = 1 | + (0.766 + 0.642i)5-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)23-s + (0.173 + 0.984i)25-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)35-s + 37-s + (0.173 − 0.984i)41-s + (0.766 + 0.642i)43-s + (−0.939 − 0.342i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)5-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)23-s + (0.173 + 0.984i)25-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)35-s + 37-s + (0.173 − 0.984i)41-s + (0.766 + 0.642i)43-s + (−0.939 − 0.342i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.585469377 + 1.242285378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.585469377 + 1.242285378i\) |
\(L(1)\) |
\(\approx\) |
\(1.437608009 + 0.3193225139i\) |
\(L(1)\) |
\(\approx\) |
\(1.437608009 + 0.3193225139i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + (0.766 + 0.642i)T \) |
| 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
−23.457542673819785939595125617758, −23.00313694909036347424327275094, −21.7366258993583810267911517259, −21.01005621404994517694351890357, −20.243047185391857943715296180124, −19.62230510717824834877062890513, −18.20034450126631835136019433554, −17.42918830261784495740188742452, −16.998782716366493564137077299681, −15.91234741445661795915724908479, −14.74474323340437179184081325007, −14.06376555522987807962108612413, −13.03578701473699624956739309532, −12.42445010899439557711844593643, −11.17063804543025633644352789947, −10.17272339883044938153456022148, −9.5382301145354938301041842214, −8.32429535132763682486834149464, −7.49279825746591799351630008477, −6.33719968759698803370856219553, −5.24104426131829081267178013531, −4.45586923674436124103981090873, −3.18361376016623147993850132826, −1.656704100301222934012693557498, −0.89857233097037794163865410333,
1.17679612168490789566685988397, 2.33259109268789141497548419386, 3.27733056413544205841788283486, 4.70949986992775859652919264453, 5.83241472749000136362849960386, 6.45018385326706630807127641447, 7.67305918682382145660996543794, 8.88810707536793821046229276655, 9.45245496489848158430624459810, 10.72531776619921729041921963655, 11.44384433676137165916262325976, 12.35589354512300621196696785702, 13.59570772250717125335676058416, 14.34209055230083683500901356381, 14.90799965501601132714415518683, 16.20946459288945596562520505251, 16.9387376313041601658496121918, 18.008902505926892587996031340036, 18.70357052757350882720055272181, 19.26054040797917587179315400112, 20.76155405717243149418736910108, 21.41082174578430940850424924306, 21.967750815137690273835016223496, 22.89446726100419590987961201604, 23.980305758281668802241377024535