L(s) = 1 | + (−0.0747 − 0.997i)5-s + (−0.365 − 0.930i)11-s + (0.988 + 0.149i)13-s + (0.222 − 0.974i)17-s + 19-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.733 − 0.680i)29-s + (−0.5 − 0.866i)31-s + (−0.222 + 0.974i)37-s + (−0.0747 − 0.997i)41-s + (−0.0747 + 0.997i)43-s + (0.365 + 0.930i)47-s + (−0.222 − 0.974i)53-s + (−0.900 + 0.433i)55-s + ⋯ |
L(s) = 1 | + (−0.0747 − 0.997i)5-s + (−0.365 − 0.930i)11-s + (0.988 + 0.149i)13-s + (0.222 − 0.974i)17-s + 19-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.733 − 0.680i)29-s + (−0.5 − 0.866i)31-s + (−0.222 + 0.974i)37-s + (−0.0747 − 0.997i)41-s + (−0.0747 + 0.997i)43-s + (0.365 + 0.930i)47-s + (−0.222 − 0.974i)53-s + (−0.900 + 0.433i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8016523672 - 1.226769145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8016523672 - 1.226769145i\) |
\(L(1)\) |
\(\approx\) |
\(1.003490294 - 0.3976464919i\) |
\(L(1)\) |
\(\approx\) |
\(1.003490294 - 0.3976464919i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.0747 - 0.997i)T \) |
| 11 | \( 1 + (-0.365 - 0.930i)T \) |
| 13 | \( 1 + (0.988 + 0.149i)T \) |
| 17 | \( 1 + (0.222 - 0.974i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.733 - 0.680i)T \) |
| 29 | \( 1 + (-0.733 - 0.680i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.222 + 0.974i)T \) |
| 41 | \( 1 + (-0.0747 - 0.997i)T \) |
| 43 | \( 1 + (-0.0747 + 0.997i)T \) |
| 47 | \( 1 + (0.365 + 0.930i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 + (0.826 + 0.563i)T \) |
| 61 | \( 1 + (0.733 + 0.680i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (-0.623 - 0.781i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.988 + 0.149i)T \) |
| 89 | \( 1 + (-0.623 - 0.781i)T \) |
| 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
−20.39588734288242883410837227197, −19.773084833861472136835419287727, −18.868637962057068597946014119693, −18.247348518447726400216281232251, −17.74310542195422407271190690598, −16.854312949380459619361475166961, −15.82451951983604071263834951481, −15.32082838822546215715336101976, −14.60007216588847777648441904816, −13.86309559074094384618255930132, −13.020992718258727662724658651, −12.30194640661844750701842056908, −11.28031726263253811752830468685, −10.762771605776456346586152693201, −10.02119332175738078991494458844, −9.19750170833825070582232375312, −8.20459263404076243283427399690, −7.34802251479551162615898923956, −6.84188294245456323040240570298, −5.79750493644772657473732449506, −5.12137165057294293602718331291, −3.76995628532709102733567509327, −3.37087973936514393905953553768, −2.21699033468736502966823498040, −1.33283300807057579111288830174,
0.570287176914690905241796816719, 1.35856554345750357226954645207, 2.65451464054544040425349342869, 3.55716826635802058984779245889, 4.44820775808596770747151729707, 5.39516479182214564532980139124, 5.87922567200364586551278229581, 7.04984676297429682263967696782, 7.94102941288072845502936209576, 8.62958326938741229440843187494, 9.28978581443425745231644023158, 10.093033458135295038157116183663, 11.334611391744548236597314383023, 11.51895263234722446796319217431, 12.66826123372357772983866651843, 13.36700238018348195649231788936, 13.81205009975674138222847910748, 14.84823464369174988332502129230, 15.95663127040711328823978753461, 16.14972473204838561165583656662, 16.9219197424572653940113192636, 17.804986115186505976847525740257, 18.69440680845197899073860959894, 19.09131053430316602697285117736, 20.29913412504959780638504014052