L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 8-s − 10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 26-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 8-s − 10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5848818461 - 0.8352978428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5848818461 - 0.8352978428i\) |
\(L(1)\) |
\(\approx\) |
\(0.8954431031 - 0.6923561364i\) |
\(L(1)\) |
\(\approx\) |
\(0.8954431031 - 0.6923561364i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + T \) |
| 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
−32.77977477191635175052479928707, −31.69105817267826320402015045987, −30.50688813647278042263812596974, −29.99619637755056753644420341532, −27.905151258129569850879188385146, −27.01938744729276436366272766231, −25.77293616548944446919539962860, −25.09940188558189703269695412665, −23.476121267368988144630363374995, −22.9077363919460848194102648851, −21.84894819157507701329106616570, −20.42652069490505187112823680086, −18.825803466462598216372321256590, −17.749162918093548281792318309656, −16.49941368005613338408600373195, −15.118774609337179397613689104199, −14.60668686839869052776940890916, −13.041768853838539638123358954091, −11.85555253194053297383153143549, −10.24074269554004572977679438718, −8.43150307697441094448459361366, −7.25306972434165787364303585032, −6.137847528495801525654894715699, −4.42515541931415033394323999481, −3.05970365132883704779328236130,
1.32821262109539135258341771297, 3.45167149451680998132885510854, 4.65583930931948334369626718444, 6.14272016679861427704041497964, 8.37098968067884270714435170930, 9.488615727841845898310548465755, 11.13785890681736891820344083328, 12.03445334948218822454021350630, 13.22572724941768578506121770247, 14.35475041691370173720337726093, 15.83741552112308547666824159107, 17.1124675864824073376317424590, 18.919270000279329932992702307033, 19.53681701562349108463893438877, 20.92813980238998848858092587711, 21.57043983198953744942275975991, 23.19708728743359872847696700867, 23.82809214140261646973097719560, 25.092307931508836039654082050908, 26.934617236798999802037220843634, 27.81713209238714104521196165823, 28.76593905865934067567157657960, 29.84452061475495958720216880350, 30.95888335854153935118747164672, 31.99243774663527132819146203647