| L(s) = 1 | − 46.7i·3-s − 255.·5-s + 3.52e3i·7-s − 2.18e3·9-s + 1.89e4i·11-s − 3.23e4·13-s + 1.19e4i·15-s + 1.13e5·17-s + 2.39e5i·19-s + 1.64e5·21-s + 2.41e4i·23-s − 3.25e5·25-s + 1.02e5i·27-s + 8.86e5·29-s + 1.22e6i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.408·5-s + 1.46i·7-s − 0.333·9-s + 1.29i·11-s − 1.13·13-s + 0.236i·15-s + 1.35·17-s + 1.83i·19-s + 0.846·21-s + 0.0862i·23-s − 0.832·25-s + 0.192i·27-s + 1.25·29-s + 1.32i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.035499422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035499422\) |
| \(L(5)\) |
|
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 | \( 1 + 255.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 3.52e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.89e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.23e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.13e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 2.39e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.41e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 8.86e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.22e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 4.96e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 4.51e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 5.43e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.25e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 5.72e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 7.88e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.02e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.56e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 4.01e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.00e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 7.07e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.02e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.39e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.42e8T + 7.83e15T^{2} \) |
| 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
−10.17244189962231020856554493330, −9.654639849961831935416626639588, −8.357365766682834777402413467949, −7.78423513274650191330444316768, −6.76294468492560377874387144586, −5.65761184496815776156397399393, −4.87984486418773258041596126683, −3.39328657116469265011333188020, −2.29693707622401239674949254894, −1.45533939369416350081958884741,
0.25188442555976552839851543584, 0.832946210659579927702441827186, 2.72411979033952007973282651295, 3.67233654779926787071999568899, 4.52656375381331009676131911875, 5.53493743165735603108574925409, 6.86356050042762816781682308443, 7.65306985591958324867041746178, 8.560265381207211020370837357704, 9.761888463514290747633948797950
