| L(s) = 1 | − 27i·3-s + 124. i·5-s − 1.16e3·7-s − 729·9-s − 2.26e3i·11-s − 3.02e3i·13-s + 3.35e3·15-s − 1.81e4·17-s + 3.32e4i·19-s + 3.14e4i·21-s − 8.58e4·23-s + 6.27e4·25-s + 1.96e4i·27-s − 1.58e5i·29-s − 1.99e5·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.443i·5-s − 1.28·7-s − 0.333·9-s − 0.513i·11-s − 0.382i·13-s + 0.256·15-s − 0.894·17-s + 1.11i·19-s + 0.741i·21-s − 1.47·23-s + 0.802·25-s + 0.192i·27-s − 1.20i·29-s − 1.20·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9799454145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9799454145\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 124. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 1.16e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.26e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 3.02e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.81e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.32e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 8.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.58e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.99e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.54e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 2.18e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.48e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 7.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.18e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.97e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 3.14e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 1.03e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.65e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.63e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 2.71e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.27e7T + 8.07e13T^{2} \) |
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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
−10.18302231143821370951307861600, −9.268043915998378948017284575643, −8.224068978176817862343824342591, −7.27196021735492916411374051187, −6.30002345998903610477344664813, −5.77955007257711156258570115703, −4.01888263212226765891767677736, −3.07964624763266414930747956905, −2.03507010399311165443689894438, −0.50394715644292722869728537621,
0.37943496423931408659194180072, 2.04735496090570305582225142491, 3.26066329124415270619376887538, 4.26795090796981257530321619439, 5.21869349893344768160326481725, 6.42529311552764507915737107257, 7.16898158661506530098335754275, 8.678778324150062133248660350038, 9.227292773970715991758020367448, 10.05134831560036476777899914743
