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 | \( 1 + (0.5 + 0.866i)T \) |
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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
−20.77214216300178674174535060307, −19.79218333866175096716957197510, −19.28669602103966645633881229797, −18.450046206036716018105761575607, −17.692789278539304869034365221683, −16.766231840557575068049146128893, −15.946702947343016627150761707527, −15.35281723618172320757819534859, −14.36668426515367051872510901044, −13.87998906714487344529685976298, −13.257533706450259496141255328046, −12.39386329740768456838740299864, −11.160045218633715803396871501692, −10.58637529627322329996088025789, −9.9088378352447527312055967030, −8.77009398095330943975232641138, −8.16396356402307127359554671323, −7.69278003376585223977523453171, −6.57732537981852090008132173670, −5.57304015324578234774301307842, −4.55196410368460148352912944086, −3.69790977155262169261702897702, −3.08573652275005701598641119434, −1.87862649459296909555114989785, −0.98916050886957563681300281388,
1.272427919782015046600019475944, 2.160733924756510596941776084804, 2.81013295728292247190136476513, 4.00574296352881125425155405560, 4.681572659631224470469136593150, 5.80283435483283978965608806131, 6.70980628362424792655300334437, 7.71295460331044037724027840044, 8.30786697887108091209733206740, 8.947002434572814568228763244873, 9.85967531279433964250678130496, 10.59882170772887525886160768857, 11.69397284280332468752063612645, 12.48319667046500826979545505040, 13.052866821031127634799329037015, 14.14560185548325691602888345235, 14.54367255320290640292660143856, 15.38286977723146298739985474160, 15.927316174754372191053681168748, 16.9865016265641307302632558705, 18.05858269802272625337310119369, 18.52675471350253332817645295941, 19.035630227741274192015370692394, 20.18633884780523919059383497631, 20.64987570210744045900333864702