| L(s) = 1 | + 46.7i·3-s + 425.·5-s + 584. i·7-s − 2.18e3·9-s − 2.42e4i·11-s − 2.33e4·13-s + 1.99e4i·15-s − 4.25e4·17-s − 1.22e5i·19-s − 2.73e4·21-s + 8.89e4i·23-s − 2.09e5·25-s − 1.02e5i·27-s − 3.33e5·29-s + 5.22e5i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.680·5-s + 0.243i·7-s − 0.333·9-s − 1.65i·11-s − 0.816·13-s + 0.393i·15-s − 0.508·17-s − 0.942i·19-s − 0.140·21-s + 0.317i·23-s − 0.536·25-s − 0.192i·27-s − 0.471·29-s + 0.566i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.617264217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617264217\) |
| \(L(5)\) |
|
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 | \( 1 - 425.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 584. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.42e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.33e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 4.25e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.22e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 8.89e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.33e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 5.22e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.76e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.48e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 5.21e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 4.82e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.03e7T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.42e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 6.91e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.84e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.13e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.39e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.00e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 3.11e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.17e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 8.92e7T + 7.83e15T^{2} \) |
| 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
−10.17801587433033007988986948017, −9.235840990312460214330495219369, −8.679577762773706762376828606508, −7.45170373843686771994767270945, −6.17607155741972872943916314899, −5.52614225677995058414396863085, −4.46951336444823846535347424596, −3.20003107570438840817653389087, −2.33407445695811922942241324008, −0.849888066607800554868102185875,
0.35398591514078553155706919488, 1.86272806712511245777696487381, 2.24688210443829857489116380200, 3.91654799267021982273828163906, 5.00846893438553574004882617240, 6.02263808511565019078319901830, 7.08321754564682801005094583361, 7.63040890629228963127873221752, 8.915747008252603389127155070399, 9.859670866358750941395199728637
