L(s) = 1 | + (−0.984 − 0.173i)3-s + (0.642 − 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.342 − 0.939i)17-s + (−0.173 + 0.984i)19-s + (−0.766 + 0.642i)21-s + (−0.866 − 0.5i)23-s + (−0.866 − 0.5i)27-s + (−0.5 − 0.866i)29-s − 31-s + (0.342 + 0.939i)33-s + (0.173 + 0.984i)39-s + (−0.939 + 0.342i)41-s + ⋯ |
L(s) = 1 | + (−0.984 − 0.173i)3-s + (0.642 − 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.342 − 0.939i)17-s + (−0.173 + 0.984i)19-s + (−0.766 + 0.642i)21-s + (−0.866 − 0.5i)23-s + (−0.866 − 0.5i)27-s + (−0.5 − 0.866i)29-s − 31-s + (0.342 + 0.939i)33-s + (0.173 + 0.984i)39-s + (−0.939 + 0.342i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01875806128 - 0.4774811751i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01875806128 - 0.4774811751i\) |
\(L(1)\) |
\(\approx\) |
\(0.6474278692 - 0.2439544307i\) |
\(L(1)\) |
\(\approx\) |
\(0.6474278692 - 0.2439544307i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.984 - 0.173i)T \) |
| 7 | \( 1 + (0.642 - 0.766i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.342 - 0.939i)T \) |
| 17 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (-0.939 + 0.342i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.642 + 0.766i)T \) |
| 59 | \( 1 + (-0.766 + 0.642i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.642 + 0.766i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.342 + 0.939i)T \) |
| 89 | \( 1 + (-0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−21.33434477211982280583500625982, −20.41418372484811105516895729708, −19.471267238209118402516323670834, −18.53110931520646765627851079150, −18.02681534638878873700662920268, −17.34259564075780073427833103968, −16.62973308315963621797000937212, −15.76517142329741139029978868885, −15.09578382005596485897190497705, −14.47835549634780466027393189481, −13.24094783516373068616894103575, −12.48172822728976422716738761330, −11.88293309748307703240530239996, −11.176634528093090306963306707599, −10.40479560944948767629205711581, −9.55722077538606176559456290214, −8.7913812274416134150104972730, −7.66292241230689439001085511493, −6.95384570060732781115678364401, −6.00967463574079959908182978019, −5.20306792903605918258330445208, −4.63347210529003384332475048940, −3.672139479828835435699204630298, −2.15799046370245356569878555307, −1.588195018878072352364653613,
0.21844958108713030469818367009, 1.16434781159124319689343621204, 2.32469468571225191442622089098, 3.60217481256395334567201600801, 4.45932750372195453867545591087, 5.47264308458956711729591092764, 5.82568003810040487086601923778, 7.0774232083045744844357860469, 7.69406890157231965096538094907, 8.38458977154856546324393063430, 9.80782831162594013141057291441, 10.42623621299346127387790296181, 11.021536902507696707570413058674, 11.84142834506520158052509348689, 12.536358537498386754771620197897, 13.46744437324733280630861741074, 14.04445536579113311916673368966, 15.06821490653787037377425473236, 15.952366433921629179772542056178, 16.71029278139783301383409364098, 17.11068390738669247348384453177, 18.24378734166656275465631925099, 18.37459753959973892737183164637, 19.44773940584612532368035667102, 20.54820124007241300971068405745