| L(s) = 1 | − 365. i·3-s + 321.·5-s − 7.74e3i·7-s − 7.47e4·9-s − 6.30e4i·11-s − 5.88e5·13-s − 1.17e5i·15-s − 1.10e6·17-s + 1.40e6i·19-s − 2.83e6·21-s + 8.10e6i·23-s − 9.66e6·25-s + 5.72e6i·27-s + 7.53e6·29-s − 3.74e7i·31-s + ⋯ |
| L(s) = 1 | − 1.50i·3-s + 0.102·5-s − 0.460i·7-s − 1.26·9-s − 0.391i·11-s − 1.58·13-s − 0.154i·15-s − 0.778·17-s + 0.568i·19-s − 0.693·21-s + 1.25i·23-s − 0.989·25-s + 0.398i·27-s + 0.367·29-s − 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.8124280536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8124280536\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 365. iT - 5.90e4T^{2} \) |
| 5 | \( 1 - 321.T + 9.76e6T^{2} \) |
| 7 | \( 1 + 7.74e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 6.30e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 5.88e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.10e6T + 2.01e12T^{2} \) |
| 19 | \( 1 - 1.40e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 8.10e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 7.53e6T + 4.20e14T^{2} \) |
| 31 | \( 1 + 3.74e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 6.74e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.94e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 3.98e7iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 3.13e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 2.99e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 9.75e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 5.95e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 1.47e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 2.76e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.82e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 3.76e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 6.30e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 6.15e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 9.73e9T + 7.37e19T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21947830389639570013291077307, −9.235569384325591677990120880171, −7.85621862731456909250220304279, −7.48260885526288032508865822370, −6.47184720514937692999495686847, −5.53934958719432870772853083283, −4.12929610710086006420413027984, −2.63357618269367576141264579246, −1.82437936928886489670488204854, −0.71532390536648814872235653240,
0.20150155986574525028147169529, 2.16607274742299021265406104231, 3.06800390470428300399366132769, 4.55012034593526439295973388164, 4.78297492836710413072800495285, 6.10414537370186644893982745498, 7.35925342695864919781705522456, 8.686081874874345696083368104870, 9.439771692886742108281790202368, 10.12610276728832722039100983589
