| L(s) = 1 | + 2·2-s + 2·4-s − 4·5-s + 6·7-s + 4·8-s − 8·10-s − 8·11-s + 14·13-s + 12·14-s + 8·16-s + 16·17-s − 6·19-s − 8·20-s − 16·22-s + 6·23-s − 4·25-s + 28·26-s + 12·28-s − 6·29-s − 8·31-s + 8·32-s + 32·34-s − 24·35-s + 12·37-s − 12·38-s − 16·40-s + 2·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.78·5-s + 2.26·7-s + 1.41·8-s − 2.52·10-s − 2.41·11-s + 3.88·13-s + 3.20·14-s + 2·16-s + 3.88·17-s − 1.37·19-s − 1.78·20-s − 3.41·22-s + 1.25·23-s − 4/5·25-s + 5.49·26-s + 2.26·28-s − 1.11·29-s − 1.43·31-s + 1.41·32-s + 5.48·34-s − 4.05·35-s + 1.97·37-s − 1.94·38-s − 2.52·40-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.733023320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.733023320\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + 4 p T^{2} + 52 T^{3} + 151 T^{4} + 52 p T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 152 T^{3} + 532 T^{4} + 152 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 132 T^{3} + 959 T^{4} + 132 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 6 T + 18 T^{2} + 36 T^{3} - 457 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 426 T^{3} + 2744 T^{4} - 426 p T^{5} + 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 148 T^{3} - 1997 T^{4} + 148 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 976 T^{3} + 7378 T^{4} - 976 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 45 T^{2} - 828 T^{3} - 9748 T^{4} - 828 p T^{5} + 45 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 24 T + 180 T^{2} + 300 T^{3} - 11209 T^{4} + 300 p T^{5} + 180 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 113 T^{2} + 726 T^{3} + 3848 T^{4} + 726 p T^{5} + 113 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 328 T^{3} - 7217 T^{4} + 328 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.180822846145119444552502177762, −7.76793829418102396565736011598, −7.54129978809340081514120468882, −7.49623789659856252394757604031, −7.29833370781179737730161232097, −7.15667263229433816755066088633, −6.40201152323026420968147094451, −5.97260544473672496512982810917, −5.82368261525715684172386052355, −5.76707694591779419618326178457, −5.59697326998474436215367664780, −5.26691718582701734904959211320, −4.95555497803790479961229531889, −4.72198208562042743276858694158, −4.35372151649427583173150479687, −4.05398443062836417607547467566, −3.88902546636047899246389647939, −3.65557597860058549489536714262, −3.32024694771716802622464338289, −3.29064747477102895528235410514, −2.60724110727375517500544306495, −2.11109531316975509083974371164, −1.55570902393976149217864041165, −1.35198816614929070096084682030, −0.861254593222668367361438668393,
0.861254593222668367361438668393, 1.35198816614929070096084682030, 1.55570902393976149217864041165, 2.11109531316975509083974371164, 2.60724110727375517500544306495, 3.29064747477102895528235410514, 3.32024694771716802622464338289, 3.65557597860058549489536714262, 3.88902546636047899246389647939, 4.05398443062836417607547467566, 4.35372151649427583173150479687, 4.72198208562042743276858694158, 4.95555497803790479961229531889, 5.26691718582701734904959211320, 5.59697326998474436215367664780, 5.76707694591779419618326178457, 5.82368261525715684172386052355, 5.97260544473672496512982810917, 6.40201152323026420968147094451, 7.15667263229433816755066088633, 7.29833370781179737730161232097, 7.49623789659856252394757604031, 7.54129978809340081514120468882, 7.76793829418102396565736011598, 8.180822846145119444552502177762
