| L(s) = 1 | − 4·2-s + 8·4-s − 4·5-s + 12·7-s − 8·8-s + 6·9-s + 16·10-s − 2·11-s + 8·13-s − 48·14-s − 4·16-s + 16·17-s − 24·18-s − 12·19-s − 32·20-s + 8·22-s − 12·23-s + 11·25-s − 32·26-s + 96·28-s − 8·31-s + 32·32-s − 64·34-s − 48·35-s + 48·36-s − 24·37-s + 48·38-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 1.78·5-s + 4.53·7-s − 2.82·8-s + 2·9-s + 5.05·10-s − 0.603·11-s + 2.21·13-s − 12.8·14-s − 16-s + 3.88·17-s − 5.65·18-s − 2.75·19-s − 7.15·20-s + 1.70·22-s − 2.50·23-s + 11/5·25-s − 6.27·26-s + 18.1·28-s − 1.43·31-s + 5.65·32-s − 10.9·34-s − 8.11·35-s + 8·36-s − 3.94·37-s + 7.78·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4740836890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4740836890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + p T^{2} - 8 T^{3} - 44 T^{4} - 8 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 300 T^{3} + 912 T^{4} - 300 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 26 T^{3} - 32 T^{4} + 26 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 97 T^{2} + 588 T^{3} + 2976 T^{4} + 588 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 9 T^{2} - 156 T^{3} - 484 T^{4} - 156 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T + 13 T^{2} - 88 T^{3} - 344 T^{4} - 88 p T^{5} + 13 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 2472 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 181 T^{2} + 1314 T^{3} + 8076 T^{4} + 1314 p T^{5} + 181 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 14 T + 65 T^{2} + 474 T^{3} - 6280 T^{4} + 474 p T^{5} + 65 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 88 p T^{5} - 19 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 1264 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 462 T^{3} + 848 T^{4} + 462 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 261 T^{2} + 2088 T^{3} + 24452 T^{4} + 2088 p T^{5} + 261 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 762 T^{3} + 2600 T^{4} + 762 p T^{5} + 65 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 22 T + 185 T^{2} - 290 T^{3} - 14024 T^{4} - 290 p T^{5} + 185 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 13094 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.775831417875831444823646892583, −9.027960482057989728412776109877, −8.763488774264092612253452501915, −8.630019775498156883545730881250, −8.621675900022574114417552356063, −8.173910521797010645516523671868, −7.991003120570099276850228682624, −7.79962256771825930189388813332, −7.79202337721623209110885083841, −7.41543808765625467287415855638, −7.32426765560973569965254989170, −6.72715406587952014398251117116, −6.64156703826392183683248293265, −5.57501254924254535274991729191, −5.55595172985257798959387442574, −5.29689421729481690546994889802, −4.50250297186734947751314730599, −4.47712495211848914475991319712, −4.22905994998182250667089184265, −3.75466490335018808154778992852, −3.52482606450636738741377164493, −1.97125040787209740677938336346, −1.75626580449775310729311826213, −1.58469767340781759911411983391, −1.13575259775052458589247932869,
1.13575259775052458589247932869, 1.58469767340781759911411983391, 1.75626580449775310729311826213, 1.97125040787209740677938336346, 3.52482606450636738741377164493, 3.75466490335018808154778992852, 4.22905994998182250667089184265, 4.47712495211848914475991319712, 4.50250297186734947751314730599, 5.29689421729481690546994889802, 5.55595172985257798959387442574, 5.57501254924254535274991729191, 6.64156703826392183683248293265, 6.72715406587952014398251117116, 7.32426765560973569965254989170, 7.41543808765625467287415855638, 7.79202337721623209110885083841, 7.79962256771825930189388813332, 7.991003120570099276850228682624, 8.173910521797010645516523671868, 8.621675900022574114417552356063, 8.630019775498156883545730881250, 8.763488774264092612253452501915, 9.027960482057989728412776109877, 9.775831417875831444823646892583
