| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 140·5-s − 216·6-s − 60·7-s + 512·8-s + 729·9-s − 1.12e3·10-s + 3.97e3·11-s − 1.72e3·12-s + 1.59e3·13-s − 480·14-s + 3.78e3·15-s + 4.09e3·16-s − 2.36e4·17-s + 5.83e3·18-s − 6.85e3·19-s − 8.96e3·20-s + 1.62e3·21-s + 3.18e4·22-s − 7.03e4·23-s − 1.38e4·24-s − 5.85e4·25-s + 1.27e4·26-s − 1.96e4·27-s − 3.84e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.500·5-s − 0.408·6-s − 0.0661·7-s + 0.353·8-s + 1/3·9-s − 0.354·10-s + 0.900·11-s − 0.288·12-s + 0.200·13-s − 0.0467·14-s + 0.289·15-s + 1/4·16-s − 1.16·17-s + 0.235·18-s − 0.229·19-s − 0.250·20-s + 0.0381·21-s + 0.636·22-s − 1.20·23-s − 0.204·24-s − 0.749·25-s + 0.142·26-s − 0.192·27-s − 0.0330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 + p^{3} T \) |
| 19 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 + 28 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 60 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3976 T + p^{7} T^{2} \) |
| 13 | \( 1 - 1592 T + p^{7} T^{2} \) |
| 17 | \( 1 + 23682 T + p^{7} T^{2} \) |
| 23 | \( 1 + 70370 T + p^{7} T^{2} \) |
| 29 | \( 1 - 97502 T + p^{7} T^{2} \) |
| 31 | \( 1 + 234890 T + p^{7} T^{2} \) |
| 37 | \( 1 - 146968 T + p^{7} T^{2} \) |
| 41 | \( 1 + 232282 T + p^{7} T^{2} \) |
| 43 | \( 1 + 409028 T + p^{7} T^{2} \) |
| 47 | \( 1 + 891710 T + p^{7} T^{2} \) |
| 53 | \( 1 - 30862 T + p^{7} T^{2} \) |
| 59 | \( 1 + 374820 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1548106 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3944928 T + p^{7} T^{2} \) |
| 71 | \( 1 - 477232 T + p^{7} T^{2} \) |
| 73 | \( 1 - 514206 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4866382 T + p^{7} T^{2} \) |
| 83 | \( 1 - 2043204 T + p^{7} T^{2} \) |
| 89 | \( 1 + 7822422 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2747090 T + p^{7} T^{2} \) |
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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
−11.70478069424092871045758471961, −11.06040326366247908053777874662, −9.729081305622310368702145217749, −8.280774078054575963366629692465, −6.90707608457718981682770998429, −6.04054158131555224612591823387, −4.61154946252590856745432307775, −3.65533615208293891543198853031, −1.79889702213183872215484773743, 0,
1.79889702213183872215484773743, 3.65533615208293891543198853031, 4.61154946252590856745432307775, 6.04054158131555224612591823387, 6.90707608457718981682770998429, 8.280774078054575963366629692465, 9.729081305622310368702145217749, 11.06040326366247908053777874662, 11.70478069424092871045758471961
