| L(s) = 1 | − 72·2-s + 2.18e3·3-s − 2.75e4·4-s − 2.21e5·5-s − 1.57e5·6-s − 2.14e6·7-s + 4.34e6·8-s + 4.78e6·9-s + 1.59e7·10-s + 3.71e7·11-s − 6.03e7·12-s − 2.79e8·13-s + 1.54e8·14-s − 4.84e8·15-s + 5.91e8·16-s + 2.49e9·17-s − 3.44e8·18-s − 4.66e9·19-s + 6.10e9·20-s − 4.69e9·21-s − 2.67e9·22-s − 1.84e10·23-s + 9.50e9·24-s + 1.85e10·25-s + 2.01e10·26-s + 1.04e10·27-s + 5.92e10·28-s + ⋯ |
| L(s) = 1 | − 0.397·2-s + 0.577·3-s − 0.841·4-s − 1.26·5-s − 0.229·6-s − 0.986·7-s + 0.732·8-s + 1/3·9-s + 0.504·10-s + 0.575·11-s − 0.486·12-s − 1.23·13-s + 0.392·14-s − 0.732·15-s + 0.550·16-s + 1.47·17-s − 0.132·18-s − 1.19·19-s + 1.06·20-s − 0.569·21-s − 0.228·22-s − 1.13·23-s + 0.422·24-s + 0.607·25-s + 0.492·26-s + 0.192·27-s + 0.830·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{7} T \) |
| 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
−22.22709361670372848431172249679, −19.70165338084625248753840444234, −18.96784897446820523487252312872, −16.63595698630902159288238337018, −14.66547827150741435452020518493, −12.53157168558172969261199283440, −9.657989850576913418310834455705, −7.81293689921521718144519552817, −3.86848774218855736996142375930, 0,
3.86848774218855736996142375930, 7.81293689921521718144519552817, 9.657989850576913418310834455705, 12.53157168558172969261199283440, 14.66547827150741435452020518493, 16.63595698630902159288238337018, 18.96784897446820523487252312872, 19.70165338084625248753840444234, 22.22709361670372848431172249679
