| L(s) = 1 | − 11.3i·2-s − 128.·4-s − 724. i·5-s − 1.95e3·7-s + 1.44e3i·8-s − 8.20e3·10-s − 2.51e4i·11-s − 6.03e3·13-s + 2.21e4i·14-s + 1.63e4·16-s − 1.46e5i·17-s − 5.95e4·19-s + 9.27e4i·20-s − 2.84e5·22-s + 1.68e5i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.15i·5-s − 0.813·7-s + 0.353i·8-s − 0.820·10-s − 1.71i·11-s − 0.211·13-s + 0.575i·14-s + 0.250·16-s − 1.75i·17-s − 0.456·19-s + 0.579i·20-s − 1.21·22-s + 0.602i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7678703061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7678703061\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 11.3iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 724. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.95e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.51e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 6.03e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.46e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 5.95e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.68e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.76e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 7.14e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 5.34e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.96e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.10e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 2.83e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 9.99e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 4.50e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.83e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.60e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.94e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.60e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 6.66e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.64e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 9.06e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.49e8T + 7.83e15T^{2} \) |
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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
−10.70356039788339454142298577120, −9.339532502749608690509125433700, −8.981492447228646401252350428798, −7.69106120974489418466390206766, −6.04056284893218608994297499551, −5.04327047038581613553020869964, −3.74253719530711999031067419731, −2.61435416719910056702796121793, −0.903684851125420247907695008225, −0.23278412989715698207983892120,
1.94660556983890851511802763683, 3.35835542406641963367878152238, 4.55709050984648091303539598539, 6.13512732845275592424781694998, 6.82619986471166055473573500320, 7.67243936745741054509346385639, 9.072974114206281231367149053055, 10.13244742549880569916706687888, 10.75766158325568226608309456974, 12.49205372750432205747510975627
