L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (−0.955 − 0.294i)11-s + (0.955 + 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 + 0.997i)26-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (−0.955 − 0.294i)11-s + (0.955 + 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 + 0.997i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.132295912 + 0.2563183997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132295912 + 0.2563183997i\) |
\(L(1)\) |
\(\approx\) |
\(0.7793629759 + 0.4671539131i\) |
\(L(1)\) |
\(\approx\) |
\(0.7793629759 + 0.4671539131i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.365 + 0.930i)T \) |
| 11 | \( 1 + (-0.955 - 0.294i)T \) |
| 13 | \( 1 + (0.955 + 0.294i)T \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.826 - 0.563i)T \) |
| 29 | \( 1 + (-0.0747 - 0.997i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.826 - 0.563i)T \) |
| 41 | \( 1 + (0.988 - 0.149i)T \) |
| 43 | \( 1 + (-0.988 - 0.149i)T \) |
| 47 | \( 1 + (0.222 + 0.974i)T \) |
| 53 | \( 1 + (-0.826 - 0.563i)T \) |
| 59 | \( 1 + (-0.623 + 0.781i)T \) |
| 61 | \( 1 + (-0.900 - 0.433i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.900 - 0.433i)T \) |
| 73 | \( 1 + (0.955 - 0.294i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.955 + 0.294i)T \) |
| 89 | \( 1 + (0.733 + 0.680i)T \) |
| 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
−23.562482385007702719644440934394, −23.105299594734756455636617627982, −21.72830985757977063197882824059, −21.2466024876846389280326898450, −20.22392016070024452488672322352, −19.839406517659747289881216837425, −18.74389479751442310442248373637, −17.910474597406292013889501030561, −17.03595891623208901471921243568, −15.782337140531416523899946000530, −15.13717807982879079356506286473, −13.74802277830164648884670260209, −13.02929926988905576951547001458, −12.46236738250936008295226448798, −11.374345663408875253725093558864, −10.602961036665868454572542944963, −9.60020479402691570896035796305, −8.553338536666383259158803133480, −7.96373203617830434565184226412, −6.17178765155261327535454204807, −5.13545516931713182498685735689, −4.30940366546403867622903323409, −3.29974572841285963538211069986, −1.98406534089324435495415195998, −0.87927448331091698686413637376,
0.37571093378805575063284728542, 2.52869247042109340705812480100, 3.630953177307524351820178079884, 4.57609284478797378523186022624, 5.93312740972929090656814458166, 6.49337496534861800684588129808, 7.73642578287805365461829627468, 8.182959375755009237484470771664, 9.514489679030318025677153256205, 10.5255852401135894714668004781, 11.54169127000911544319604756696, 12.648629402089821153247791556527, 13.75853557494317054111161854413, 14.22885328578308070200918620488, 15.38860880813427876521658399574, 15.884152880987286341371232330779, 16.75059360273982361354422502721, 18.08920920115111539728031196106, 18.40407211686006416453052632979, 19.26528655699197540990270080508, 20.76573269523989820212875771225, 21.45912888913444807527216148968, 22.6052289376681093297592314179, 23.05129877288982679404017677969, 23.79429149799859046009647958598