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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1609175992 + 0.9126090552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1609175992 + 0.9126090552i\) |
\(L(1)\) |
\(\approx\) |
\(0.6064060984 + 0.5088351335i\) |
\(L(1)\) |
\(\approx\) |
\(0.6064060984 + 0.5088351335i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−31.61924848461166507709057859695, −30.1914730423111638855293434465, −29.297341620179791481871767341083, −28.35107756861216251490525551465, −27.452585828236839116525043623220, −26.19258666981815687536306759032, −25.11636804269155075797828905573, −23.778619796475470875047435601147, −22.23633508935254910997470874787, −21.188605522800932291959603082685, −20.36596382861515845328735074954, −19.193747007396696851566620539335, −17.90876090382346868774729547969, −16.99876365587067851414758810634, −15.72649471839827997692803976964, −13.53927064785408013958605420191, −12.95087453247031168489827550594, −11.44895706453950623490206758788, −10.26400469809790399446654622119, −8.95271660885719401533689139493, −8.01548491746004374237530247642, −5.75753200078613424618053984761, −4.09945934075237740670677833466, −2.3056656336572927153605874930, −0.56929975421934199570280100575,
2.066485422968420720810869899729, 4.50636601233958095101447187123, 6.21908635095735866533630495706, 7.08118031881071500841259294816, 8.64894575264804548779482324867, 9.96031711711280314728317536698, 10.99621633307619684610401204736, 13.12793947599152688345099822989, 14.35784051823042400781620578528, 15.26660074020311617709482000335, 16.57381701716998256718886067077, 17.87451831229819133540196833690, 18.50654739504890602234359164559, 19.85463657287545624233967352120, 21.51076552900045323627886547966, 22.76492095257696082177378474635, 23.691220068777317727996842046732, 25.044141992410827421985443713078, 26.01493640650596700618066475321, 26.637322351001848101011857305252, 28.10699326839897084807315874411, 28.98637124911945001210240013104, 30.47149947748832369198563694883, 31.58553865087248007431878472696, 33.04505395566049150703762732496