| L(s) = 1 | + 72·2-s − 2.75e4·4-s + 2.21e5·5-s − 2.14e6·7-s − 4.34e6·8-s + 1.59e7·10-s − 3.71e7·11-s − 2.79e8·13-s − 1.54e8·14-s + 5.91e8·16-s − 2.49e9·17-s − 4.66e9·19-s − 6.10e9·20-s − 2.67e9·22-s + 1.84e10·23-s + 1.85e10·25-s − 2.01e10·26-s + 5.92e10·28-s + 1.15e11·29-s − 5.61e10·31-s + 1.84e11·32-s − 1.79e11·34-s − 4.75e11·35-s + 6.14e11·37-s − 3.36e11·38-s − 9.62e11·40-s − 5.49e11·41-s + ⋯ |
| L(s) = 1 | + 0.397·2-s − 0.841·4-s + 1.26·5-s − 0.986·7-s − 0.732·8-s + 0.504·10-s − 0.575·11-s − 1.23·13-s − 0.392·14-s + 0.550·16-s − 1.47·17-s − 1.19·19-s − 1.06·20-s − 0.228·22-s + 1.13·23-s + 0.607·25-s − 0.492·26-s + 0.830·28-s + 1.24·29-s − 0.366·31-s + 0.951·32-s − 0.586·34-s − 1.25·35-s + 1.06·37-s − 0.476·38-s − 0.928·40-s − 0.440·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 9 p^{3} T + p^{15} T^{2} \) |
| 5 | \( 1 - 44298 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 307000 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 37169316 T + p^{15} T^{2} \) |
| 13 | \( 1 + 21536482 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 2492912754 T + p^{15} T^{2} \) |
| 19 | \( 1 + 4669782244 T + p^{15} T^{2} \) |
| 23 | \( 1 - 18467933400 T + p^{15} T^{2} \) |
| 29 | \( 1 - 115953449418 T + p^{15} T^{2} \) |
| 31 | \( 1 + 56187023200 T + p^{15} T^{2} \) |
| 37 | \( 1 - 614764926830 T + p^{15} T^{2} \) |
| 41 | \( 1 + 549859792410 T + p^{15} T^{2} \) |
| 43 | \( 1 + 982884444028 T + p^{15} T^{2} \) |
| 47 | \( 1 + 2076144322896 T + p^{15} T^{2} \) |
| 53 | \( 1 - 12048378188130 T + p^{15} T^{2} \) |
| 59 | \( 1 + 23087905758324 T + p^{15} T^{2} \) |
| 61 | \( 1 + 8505809142442 T + p^{15} T^{2} \) |
| 67 | \( 1 + 12331010771476 T + p^{15} T^{2} \) |
| 71 | \( 1 + 58989192692472 T + p^{15} T^{2} \) |
| 73 | \( 1 + 5609828808070 T + p^{15} T^{2} \) |
| 79 | \( 1 - 159918683826800 T + p^{15} T^{2} \) |
| 83 | \( 1 + 57675894342876 T + p^{15} T^{2} \) |
| 89 | \( 1 - 362287610413974 T + p^{15} T^{2} \) |
| 97 | \( 1 + 539786645144926 T + p^{15} T^{2} \) |
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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
−16.99984680014534843699969035970, −14.99927171164755678727077206534, −13.49112175253532971202930733528, −12.78046783673710631410236291931, −10.16206497294387033133027622092, −9.036928043013678397623611950516, −6.39339802751720463839740997055, −4.81782426540620200811285083493, −2.59551315533135264211852882595, 0,
2.59551315533135264211852882595, 4.81782426540620200811285083493, 6.39339802751720463839740997055, 9.036928043013678397623611950516, 10.16206497294387033133027622092, 12.78046783673710631410236291931, 13.49112175253532971202930733528, 14.99927171164755678727077206534, 16.99984680014534843699969035970
