| L(s) = 1 | + 46.7i·3-s − 317.·5-s + 2.58e3i·7-s − 2.18e3·9-s + 2.49e4i·11-s − 2.26e4·13-s − 1.48e4i·15-s − 8.34e4·17-s + 1.31e5i·19-s − 1.20e5·21-s − 6.29e3i·23-s − 2.89e5·25-s − 1.02e5i·27-s − 4.48e5·29-s + 8.48e5i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.508·5-s + 1.07i·7-s − 0.333·9-s + 1.70i·11-s − 0.794·13-s − 0.293i·15-s − 0.999·17-s + 1.00i·19-s − 0.621·21-s − 0.0225i·23-s − 0.741·25-s − 0.192i·27-s − 0.633·29-s + 0.919i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7711591033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7711591033\) |
| \(L(5)\) |
|
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 | \( 1 + 317.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 2.58e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.49e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.26e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 8.34e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.31e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 6.29e3iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.48e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 8.48e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 6.12e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 7.01e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 5.33e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 9.53e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 5.69e5T + 6.22e13T^{2} \) |
| 59 | \( 1 - 8.97e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.79e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.97e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 2.29e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.29e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.82e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 8.08e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 6.45e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 2.32e7T + 7.83e15T^{2} \) |
| 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
−10.53706356429038300462105348830, −9.682719610129212805977569636977, −9.011011323854129183054825312974, −7.926280648484929512983581630542, −7.03507686183036494891624826672, −5.79678850567747872612213492381, −4.80023024293871871830529569741, −4.04389456275490895685575731615, −2.65945419374476446291212412955, −1.78612012578268385096851810786,
0.23636065767927780685155427571, 0.58599005276806952163390801625, 2.11680782359932014249366642606, 3.33901066698128805828071787802, 4.26561643667630644015528437248, 5.52448710218181450187380015914, 6.65885366599461758860942867990, 7.39273074202203126035138971873, 8.247911741035835816128919597342, 9.139148922508205019554342612609
