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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.078524845 - 0.6803642328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078524845 - 0.6803642328i\) |
\(L(1)\) |
\(\approx\) |
\(0.9293177395 - 0.4665857671i\) |
\(L(1)\) |
\(\approx\) |
\(0.9293177395 - 0.4665857671i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−24.418381893326993251195928447534, −23.24354372489981488773234938346, −22.70526164001530358811074749812, −21.95645178592288253164888113028, −20.93422708756531115840671396228, −19.58339668202495831109083008562, −18.81114986812252541340792824400, −18.09150084431808470046543125875, −17.28873665700007477986225749137, −16.46196663261835381048499567624, −15.37120640112898771392009401650, −14.81398087534774868559543323533, −13.74196071822504258471463830666, −13.345846640727924437079218069460, −11.69043763841102785388214023393, −10.767619732777142823966573097465, −9.763292969865773640374025212022, −8.918892343838653573301299715324, −7.93859901131752862844125200160, −6.65090151126309694596765505690, −6.44210802279498852384952981548, −5.16132328865290821377063771287, −3.97857029674979354240196885067, −2.76635275768859924235028729082, −1.03366962718955399239824979823,
1.21472431801784324147653515687, 1.84358319565128165090325466849, 3.505932728850079756063270884819, 4.26696368234296859117300429063, 5.376437156495071812024686563241, 6.54775677078679872297989446738, 8.17773106281963165909456333461, 8.7736052117806921912449176225, 9.62854522613829332030980388839, 10.547079099877621190693270327734, 11.60196099530654554216614179644, 12.39829888486255331624711052342, 13.159936361413258831003855506286, 13.972608351670297634021536297583, 15.055246969990743015874145711307, 16.55524735250601949635086236104, 17.00176905750009085223234875418, 17.92074471841603834017884671591, 18.89815262671151139017159893915, 19.75095833284061194074207929952, 20.43945209798434395299306046296, 21.3263978113851914047230815959, 21.79266692307006839010796340885, 23.08009382739730833072429885240, 23.61745447811989393191475195093