| L(s) = 1 | − 585. i·5-s − 1.14e3·7-s − 1.59e4i·11-s − 2.73e4·13-s + 8.54e4i·17-s + 1.58e5·19-s + 1.21e5i·23-s + 4.78e4·25-s + 4.85e5i·29-s + 1.70e6·31-s + 6.71e5i·35-s − 9.33e5·37-s − 1.06e6i·41-s + 4.97e6·43-s − 3.79e6i·47-s + ⋯ |
| L(s) = 1 | − 0.936i·5-s − 0.477·7-s − 1.09i·11-s − 0.957·13-s + 1.02i·17-s + 1.21·19-s + 0.432i·23-s + 0.122·25-s + 0.686i·29-s + 1.85·31-s + 0.447i·35-s − 0.498·37-s − 0.377i·41-s + 1.45·43-s − 0.776i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.835734831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835734831\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 585. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.14e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.59e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.73e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 8.54e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.58e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.21e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 4.85e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.70e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 9.33e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.06e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 4.97e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.79e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.30e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.80e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.37e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.68e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.32e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.78e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 8.40e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 7.57e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 9.77e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.41e8T + 7.83e15T^{2} \) |
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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
−9.533576735229882086878575913051, −8.692142881787789397252815994483, −7.929291505962450564633974117533, −6.78531816708961579508742840190, −5.70901059437002304761782532977, −4.95084944478962374017500196438, −3.75587312417829172448421883249, −2.74787599779755130217656504368, −1.29497955425419556181763361177, −0.45909837133092762552007705618,
0.823195879386526408441139908838, 2.41982571452337638478101020589, 2.95849685719241465465017775645, 4.34115278185972574639838365339, 5.28896174731493124549619511705, 6.63044578211046761469007793325, 7.12649477402029049264471930236, 8.024290444949303883854044350006, 9.666618930005794246387115408469, 9.721678696206396722538404496836
