| L(s) = 1 | − 46.7i·3-s + 149.·5-s − 3.20e3i·7-s − 2.18e3·9-s − 3.23e3i·11-s − 176.·13-s − 6.98e3i·15-s + 8.04e4·17-s − 2.39e5i·19-s − 1.50e5·21-s − 8.53e4i·23-s − 3.68e5·25-s + 1.02e5i·27-s + 3.10e5·29-s + 4.94e5i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.238·5-s − 1.33i·7-s − 0.333·9-s − 0.220i·11-s − 0.00618·13-s − 0.137i·15-s + 0.963·17-s − 1.83i·19-s − 0.771·21-s − 0.304i·23-s − 0.942·25-s + 0.192i·27-s + 0.438·29-s + 0.535i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.457296654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457296654\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| good | 5 | \( 1 - 149.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 3.20e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 3.23e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 176.T + 8.15e8T^{2} \) |
| 17 | \( 1 - 8.04e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.39e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 8.53e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 3.10e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 4.94e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 8.55e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.41e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 2.15e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 1.10e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.47e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.62e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 9.65e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.25e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.04e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.01e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.96e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 4.44e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.11e8T + 3.93e15T^{2} \) |
| 97 | \( 1 - 3.96e6T + 7.83e15T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61060654342331865837115185403, −9.660107772692137153962546178274, −8.411693774548415863597285328595, −7.35982923411831833241512966206, −6.65331631619584571857750740137, −5.32005427893811203891815303254, −4.03366039442386837340705948442, −2.73499754409042352389481315307, −1.24874544429220755576545746284, −0.34776173807277469841245632113,
1.59079465831227328734475564940, 2.81427537888488039488729688397, 4.04755181284098766836973355695, 5.50072546949588278879159176473, 5.98226752805559162119398486215, 7.70389141813213256569150306154, 8.657289876618764069965169514690, 9.675872307128389903324320288895, 10.31779293911077793850648987679, 11.79210498736156902238072281864
