| L(s) = 1 | − 27i·3-s − 8.99i·5-s − 616.·7-s − 729·9-s + 3.03e3i·11-s − 1.22e4i·13-s − 242.·15-s − 3.54e4·17-s − 4.68e4i·19-s + 1.66e4i·21-s + 8.98e4·23-s + 7.80e4·25-s + 1.96e4i·27-s + 1.13e5i·29-s − 2.46e5·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.0321i·5-s − 0.679·7-s − 0.333·9-s + 0.687i·11-s − 1.54i·13-s − 0.0185·15-s − 1.75·17-s − 1.56i·19-s + 0.392i·21-s + 1.53·23-s + 0.998·25-s + 0.192i·27-s + 0.861i·29-s − 1.48·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2975413612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2975413612\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
| good | 5 | \( 1 + 8.99iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 616.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.03e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 1.22e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.54e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.68e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 8.98e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.13e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.46e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.83e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.38e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.97e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 6.65e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.02e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.66e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.94e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 5.09e4iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.04e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.82e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.30e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.20e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 5.21e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.08e6T + 8.07e13T^{2} \) |
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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
−10.65164418869485528310269964115, −9.173822710036181159217843710371, −8.772115905807918548258706872663, −7.16697289677321064347601839716, −7.00001612024124397726942922924, −5.62435338164613987518900069384, −4.67074008551684259412619073203, −3.17114732840223245895533223851, −2.34574337203488733611275729664, −0.854642493021223640373072558724,
0.07381753002903350747343313731, 1.67702476505267422827897945120, 3.00444880983487775958517254363, 3.99252023549375528966589695919, 4.94078459597884724859708887663, 6.26680090590678999314939200211, 6.84334922886902708014817526212, 8.323226893748904222895492193913, 9.132779559192068369322812759449, 9.764746085379385825549764645823
