L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.866 − 0.5i)7-s + (0.5 − 0.866i)9-s + 11-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 + 0.866i)21-s − 23-s − i·27-s − i·29-s + i·31-s + (−0.866 + 0.5i)33-s + (0.866 + 0.5i)39-s + (0.5 + 0.866i)41-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.866 − 0.5i)7-s + (0.5 − 0.866i)9-s + 11-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 + 0.866i)21-s − 23-s − i·27-s − i·29-s + i·31-s + (−0.866 + 0.5i)33-s + (0.866 + 0.5i)39-s + (0.5 + 0.866i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9462058946 + 0.5861922811i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9462058946 + 0.5861922811i\) |
\(L(1)\) |
\(\approx\) |
\(0.8681919168 + 0.1571736207i\) |
\(L(1)\) |
\(\approx\) |
\(0.8681919168 + 0.1571736207i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - iT \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.866 - 0.5i)T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (-0.866 + 0.5i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (0.866 - 0.5i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + T \) |
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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
−20.58472638144025691495910397861, −19.57686979725719601329319549436, −18.9395050308703050763927287530, −18.25058790713157799912673704596, −17.34127853848054304766832735003, −17.13061915046628068152328122579, −16.08916219369545679393288831017, −15.32270248650487781742908638193, −14.32534716040882809152113419601, −13.79139455707899594842383530539, −12.75086376847266943323412359788, −11.864363574226016165762081542977, −11.58665292322563266355169884631, −10.83821027764732239824748055124, −9.69082030172493258831315520311, −8.94872679421721927114493799321, −7.96548512777495745083691470853, −7.141749445614846207888848694724, −6.38290103352370076933147203120, −5.64912122574971553751485165535, −4.5970978718681409029213376954, −4.16308146586077126920921574644, −2.303470658340772938148607093127, −1.91083516132012593238368297168, −0.55941319275484719625153865847,
1.005935546903729650533749885892, 1.89638777395573506567987561222, 3.44099262989309695004083833923, 4.24468219850772404777251345212, 4.84160434646301283744337261170, 5.87936989977165038847400300455, 6.512925846072380977421455565875, 7.5151253195395432058388186972, 8.39913918106312198537069590367, 9.29652704158481236298185668850, 10.35856596632325731853916698054, 10.690869771018431765295389911989, 11.55064190482303490947319913107, 12.34151531008797659989503989962, 12.94911064853621863223424549758, 14.3702692111173767425548158342, 14.61010575112066147569960427642, 15.57472429075989509927698945774, 16.40798843078786942941965425954, 17.189998406985718400348519176197, 17.58715330380356884644939784423, 18.22834590717909734075554188198, 19.490884367463015056498028797720, 20.033286584435161665889184470959, 20.931504530516929694846963842892