L(s) = 1 | + (0.984 + 0.173i)3-s + (0.642 − 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)17-s + (−0.984 − 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.5 + 0.866i)23-s + (0.866 + 0.5i)27-s + (0.866 − 0.5i)29-s + i·31-s + (−0.342 − 0.939i)33-s + (0.984 − 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)3-s + (0.642 − 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)17-s + (−0.984 − 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.5 + 0.866i)23-s + (0.866 + 0.5i)27-s + (0.866 − 0.5i)29-s + i·31-s + (−0.342 − 0.939i)33-s + (0.984 − 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.560820136 - 0.5970402781i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.560820136 - 0.5970402781i\) |
\(L(1)\) |
\(\approx\) |
\(1.654403103 - 0.1496897511i\) |
\(L(1)\) |
\(\approx\) |
\(1.654403103 - 0.1496897511i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (0.984 + 0.173i)T \) |
| 7 | \( 1 + (0.642 - 0.766i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.939 - 0.342i)T \) |
| 17 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.984 - 0.173i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.642 - 0.766i)T \) |
| 59 | \( 1 + (-0.642 - 0.766i)T \) |
| 61 | \( 1 + (0.342 + 0.939i)T \) |
| 67 | \( 1 + (0.642 - 0.766i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.642 + 0.766i)T \) |
| 83 | \( 1 + (-0.342 + 0.939i)T \) |
| 89 | \( 1 + (-0.642 - 0.766i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.64987570210744045900333864702, −20.18633884780523919059383497631, −19.035630227741274192015370692394, −18.52675471350253332817645295941, −18.05858269802272625337310119369, −16.9865016265641307302632558705, −15.927316174754372191053681168748, −15.38286977723146298739985474160, −14.54367255320290640292660143856, −14.14560185548325691602888345235, −13.052866821031127634799329037015, −12.48319667046500826979545505040, −11.69397284280332468752063612645, −10.59882170772887525886160768857, −9.85967531279433964250678130496, −8.947002434572814568228763244873, −8.30786697887108091209733206740, −7.71295460331044037724027840044, −6.70980628362424792655300334437, −5.80283435483283978965608806131, −4.681572659631224470469136593150, −4.00574296352881125425155405560, −2.81013295728292247190136476513, −2.160733924756510596941776084804, −1.272427919782015046600019475944,
0.98916050886957563681300281388, 1.87862649459296909555114989785, 3.08573652275005701598641119434, 3.69790977155262169261702897702, 4.55196410368460148352912944086, 5.57304015324578234774301307842, 6.57732537981852090008132173670, 7.69278003376585223977523453171, 8.16396356402307127359554671323, 8.77009398095330943975232641138, 9.9088378352447527312055967030, 10.58637529627322329996088025789, 11.160045218633715803396871501692, 12.39386329740768456838740299864, 13.257533706450259496141255328046, 13.87998906714487344529685976298, 14.36668426515367051872510901044, 15.35281723618172320757819534859, 15.946702947343016627150761707527, 16.766231840557575068049146128893, 17.692789278539304869034365221683, 18.450046206036716018105761575607, 19.28669602103966645633881229797, 19.79218333866175096716957197510, 20.77214216300178674174535060307