| L(s) = 1 | + 46.7i·3-s + 1.08e3·5-s − 426. i·7-s − 2.18e3·9-s − 1.42e4i·11-s − 3.42e4·13-s + 5.07e4i·15-s + 2.00e4·17-s + 1.96e5i·19-s + 1.99e4·21-s + 3.47e5i·23-s + 7.85e5·25-s − 1.02e5i·27-s − 1.00e6·29-s + 1.63e6i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.73·5-s − 0.177i·7-s − 0.333·9-s − 0.972i·11-s − 1.19·13-s + 1.00i·15-s + 0.240·17-s + 1.50i·19-s + 0.102·21-s + 1.24i·23-s + 2.01·25-s − 0.192i·27-s − 1.41·29-s + 1.77i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.518623523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.518623523\) |
| \(L(5)\) |
|
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 | \( 1 - 1.08e3T + 3.90e5T^{2} \) |
| 7 | \( 1 + 426. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.42e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.42e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 2.00e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.96e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 3.47e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.00e6T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.63e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 7.91e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.36e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.50e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 1.49e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.94e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 8.50e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.78e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.49e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.84e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.11e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 3.67e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.94e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.70e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.26e8T + 7.83e15T^{2} \) |
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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
−11.05306139204808692579243420425, −10.06067041953925074004428417668, −9.626398969845603866847308305355, −8.549674946298829889350263968434, −7.12401087972544295632408571539, −5.71338565888636397984931254552, −5.38535395712677001240005840583, −3.68949882137309073259797639560, −2.43047941327878887063326314112, −1.25059674180084985943678405595,
0.56177992213494376309296461764, 2.11894964201724500502302952162, 2.44925054470023559223991053988, 4.67940846284500972896737127529, 5.65492100922759832823071253105, 6.65958740605970464534600683504, 7.55197640125890705974974998999, 9.137630839435625969443693339922, 9.647669269304853471068082791840, 10.67003694022040941302748093943
