L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s − 12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)18-s + (−0.309 + 0.951i)19-s + (0.809 + 0.587i)20-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s − 12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)18-s + (−0.309 + 0.951i)19-s + (0.809 + 0.587i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.143628323 + 0.2080575902i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.143628323 + 0.2080575902i\) |
\(L(1)\) |
\(\approx\) |
\(0.8408689849 + 0.03850472566i\) |
\(L(1)\) |
\(\approx\) |
\(0.8408689849 + 0.03850472566i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 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 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 + T \) |
| 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 | \( 1 - T \) |
| 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
−30.52162968969400477537506005998, −29.776205668248861807611844847938, −28.709645766484904512992613084437, −27.88679535796053365562136257351, −26.338841269895710999083861130384, −25.49639112568298736627136228352, −24.75717159691271969661857980112, −23.52360036155062422879406979006, −22.65770363604683125422513179021, −20.97719044652674181344707567032, −19.42426177922738941617704658238, −18.62654594466980353362507434157, −17.64346567685946162044269023725, −16.99195253874380967408532151966, −15.37013233647905810539566890306, −14.127959091358391932630238962747, −13.08172386382919047117883274058, −11.285532789887579610731650655534, −10.3121140632686224470846463879, −8.79579719187064015022848482775, −7.52107039970816027673152663554, −6.3975435696178778641933436552, −5.54590395183133903701703137792, −2.48212295941046432136210154003, −0.96465532198887001435257401458,
1.23182753820516844374618733760, 3.17819665792546424090336319627, 4.75190656237086900072548356975, 6.341183253011033851140828684001, 8.401848633306559213227354425640, 9.389573118774003644612996694402, 10.2713417323658349602893617409, 11.44798993251795443306792662152, 12.68221218349272029227328242458, 14.205410670274665819532065676107, 16.014550602416313814098008659110, 16.700885711614448760375008973593, 17.6793315724272640203010818700, 18.90054064037993331690103618599, 20.515395486565823834446005700496, 20.992412110723878767450417264936, 21.90421611804432634331539913870, 23.263070558992003163293086773461, 25.09799166618416766819208327394, 25.831710269202203228206864145134, 27.04661599863471794595195682540, 27.88174661697620142219442362486, 28.82303449247007220836582865578, 29.48067417311086922517288406028, 31.0220350705373066737310801196