| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 2·7-s + 3·9-s − 2·10-s + 2·12-s − 4·13-s + 4·14-s − 15-s + 2·17-s + 6·18-s − 2·20-s + 2·21-s − 4·23-s + 5·25-s − 8·26-s + 4·28-s − 2·30-s − 7·31-s + 8·32-s + 4·34-s − 2·35-s + 6·36-s − 3·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.755·7-s + 9-s − 0.632·10-s + 0.577·12-s − 1.10·13-s + 1.06·14-s − 0.258·15-s + 0.485·17-s + 1.41·18-s − 0.447·20-s + 0.436·21-s − 0.834·23-s + 25-s − 1.56·26-s + 0.755·28-s − 0.365·30-s − 1.25·31-s + 1.41·32-s + 0.685·34-s − 0.338·35-s + 36-s − 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.680758127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.680758127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T + p T^{2} - p^{2} T^{4} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 9 p T^{5} - 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 20 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 40 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 60 T^{3} + 101 T^{4} + 60 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 91 p T^{5} + 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 195 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 8 T + 23 T^{2} + 144 T^{3} - 2095 T^{4} + 144 p T^{5} + 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 240 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 420 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 5 T - 34 T^{2} - 465 T^{3} - 319 T^{4} - 465 p T^{5} - 34 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 12 T + 83 T^{2} + 264 T^{3} - 1895 T^{4} + 264 p T^{5} + 83 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 4 T - 57 T^{2} - 520 T^{3} + 2081 T^{4} - 520 p T^{5} - 57 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 580 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 780 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 1015 p T^{5} - 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04544835262353799011237969370, −9.740543239102578761272025647524, −9.139759794709922946000237174347, −9.112210136421483920628910867904, −9.022431459272493615384257859535, −8.392913923197364381012031015972, −8.135308842563757566610370447889, −7.68825012170275521608255626985, −7.65732721530845507138143271324, −7.47374844699451454782707731460, −7.07695754548104915286269228403, −6.53579163249862800585978382416, −6.26466009192094454467164952579, −6.17106539018785793875997065933, −5.58078696105245909706459075932, −5.09709042237607286669460499736, −4.97661972631937614006571407450, −4.56590598466553669105164376893, −4.52694870517532000864482317088, −3.72264831966698349248061952962, −3.48798401536756052280755262332, −3.32397112663022985450592151535, −2.65040383558416204162305663724, −2.07463231164244551369690537124, −1.54234294181023923102381425586,
1.54234294181023923102381425586, 2.07463231164244551369690537124, 2.65040383558416204162305663724, 3.32397112663022985450592151535, 3.48798401536756052280755262332, 3.72264831966698349248061952962, 4.52694870517532000864482317088, 4.56590598466553669105164376893, 4.97661972631937614006571407450, 5.09709042237607286669460499736, 5.58078696105245909706459075932, 6.17106539018785793875997065933, 6.26466009192094454467164952579, 6.53579163249862800585978382416, 7.07695754548104915286269228403, 7.47374844699451454782707731460, 7.65732721530845507138143271324, 7.68825012170275521608255626985, 8.135308842563757566610370447889, 8.392913923197364381012031015972, 9.022431459272493615384257859535, 9.112210136421483920628910867904, 9.139759794709922946000237174347, 9.740543239102578761272025647524, 10.04544835262353799011237969370
