L(s) = 1 | + (0.669 + 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 + 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)24-s + (−0.913 + 0.406i)26-s + (0.309 + 0.951i)27-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 + 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)24-s + (−0.913 + 0.406i)26-s + (0.309 + 0.951i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.075630687 + 2.077153065i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075630687 + 2.077153065i\) |
\(L(1)\) |
\(\approx\) |
\(1.394689713 + 1.187441043i\) |
\(L(1)\) |
\(\approx\) |
\(1.394689713 + 1.187441043i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.669 + 0.743i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.669 + 0.743i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.978 - 0.207i)T \) |
| 37 | \( 1 + (-0.913 + 0.406i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.104 - 0.994i)T \) |
| 53 | \( 1 + (0.978 - 0.207i)T \) |
| 59 | \( 1 + (0.104 - 0.994i)T \) |
| 61 | \( 1 + (-0.978 - 0.207i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.104 - 0.994i)T \) |
| 79 | \( 1 + (-0.669 - 0.743i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−24.33830474761730407282519922312, −23.13888613117048250809751009367, −22.56023015776697508563227714551, −21.29439501046801606054396337804, −20.77177455791703295123420925241, −19.85188749994712622702426389181, −19.28344001182769227567829462391, −18.34742345361578087554394683203, −17.487433597743006738350623960528, −15.71356546166784676192454163564, −15.196145270374797723237016827060, −14.086705173024383185149576229126, −13.57118008563319331496198408468, −12.57780475547284727635421076139, −11.911621855149795354764987378370, −10.60863367985091548914641071657, −9.762036977532678771636687454767, −8.81022976697366592380899221348, −7.66482264643768595328145972448, −6.557842170839595769750349961684, −5.36924006411536226984824869892, −4.18262560895969405889501320275, −3.14549647200161855716361856485, −2.33536867713336696055385883427, −1.069411953916988035001462743298,
2.145767039650005711225065516595, 3.14728250583433817879937766540, 4.32463908641262901725287424491, 4.89374600482684347005952619606, 6.48290584631673047173496553157, 7.18412737174909503680495670917, 8.48497066785604318275304791415, 8.89709554126472861333569443077, 10.20550237120532858793594336743, 11.44125474455863801681211301298, 12.61728317624510240761520676213, 13.48167296501703962252450519795, 14.20851014941001981148171308150, 15.05379621339457235699610280809, 15.71032716976780743026201690399, 16.647956878041635780944029065, 17.5108426724568971085096826451, 18.76392397753453074092497465340, 19.70684406260260354097985957580, 20.6471182121541163038801278313, 21.5702466983105998747698453779, 21.988499415363277908625978967282, 23.15020884670600359222323881583, 24.20584381329760996437449059279, 24.64547451290441887561548251359