L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + 7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.309 + 0.951i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s − 18-s + (0.809 − 0.587i)21-s + (0.809 − 0.587i)22-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + 7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.309 + 0.951i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s − 18-s + (0.809 − 0.587i)21-s + (0.809 − 0.587i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.255946592 - 0.8931924422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.255946592 - 0.8931924422i\) |
\(L(1)\) |
\(\approx\) |
\(1.181872211 - 0.5561468477i\) |
\(L(1)\) |
\(\approx\) |
\(1.181872211 - 0.5561468477i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (0.309 + 0.951i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.809 + 0.587i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.809 + 0.587i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.809 - 0.587i)T \) |
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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
−24.221287338044133077798448481419, −22.87779441234076145402974960821, −22.01867615342799171747685631536, −21.25950109650974484738826922869, −20.18617857972786530258946696897, −19.42651551613466713257435321918, −18.52659547518765066587814491729, −17.54305226562171837188646276339, −16.81574148689091066378694464078, −15.76697181023417185346222014646, −15.15679698479836152315420096401, −14.32538275502634987943023989300, −13.774041993822392659273902689268, −12.664180805507212641597804528851, −10.95982336922846269108071639021, −10.43037478177512736601300897020, −9.20040982389220244734776413649, −8.409799989186739761093083810347, −7.97994243670475504541100627546, −6.74619063509837604280869257116, −5.500018960898814269448369293841, −4.674926882549456824462475663160, −3.69572006379579188004675137762, −2.2380073638056415134616706483, −0.73338035817111409907603985933,
1.12106712886184222861401400652, 1.93342810841458579763198261238, 2.7836126015990060979147974271, 4.16460691503901832932441271768, 4.82961249938298601436817680101, 6.7658465134334638985597385936, 7.60466848020353811269323936481, 8.52962250123063921669261812930, 9.29609691606150841840458411375, 10.12567510341668104993170738696, 11.58279327523873590925686036061, 11.8698057914314517525604757843, 13.04013976865717165506839286382, 13.86598224354901349619052105114, 14.512279703216446491927127404832, 15.563267267218874039512539534524, 17.15653553869320127511473443971, 17.75527108115511894747482820140, 18.4515858078812717067635982108, 19.48225515966410232448120236052, 19.936520605276152073623720052643, 20.96512672120621361876529431489, 21.32972600552933002578299309070, 22.57908393376636400844256059180, 23.507427275370371666863597688176