| L(s) = 1 | + 6·2-s + 76·4-s − 688·7-s + 888·8-s + 5.10e3·9-s − 4.12e3·14-s + 1.46e4·16-s + 1.45e3·17-s + 3.06e4·18-s − 1.29e3·23-s + 2.14e5·25-s − 5.22e4·28-s − 8.92e4·31-s + 2.66e5·32-s + 8.71e3·34-s + 3.87e5·36-s + 5.21e5·41-s − 7.77e3·46-s + 1.56e6·47-s − 2.48e6·49-s + 1.28e6·50-s − 6.10e5·56-s − 5.35e5·62-s − 3.51e6·63-s + 1.23e6·64-s + 1.10e5·68-s − 7.59e6·71-s + ⋯ |
| L(s) = 1 | + 0.530·2-s + 0.593·4-s − 0.758·7-s + 0.613·8-s + 2.33·9-s − 0.402·14-s + 0.893·16-s + 0.0716·17-s + 1.23·18-s − 0.0222·23-s + 2.74·25-s − 0.450·28-s − 0.538·31-s + 1.44·32-s + 0.0380·34-s + 1.38·36-s + 1.18·41-s − 0.0117·46-s + 2.20·47-s − 3.02·49-s + 1.45·50-s − 0.464·56-s − 0.285·62-s − 1.76·63-s + 0.591·64-s + 0.0425·68-s − 2.51·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.244248455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.244248455\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 3 p T - 5 p^{3} T^{2} - 3 p^{6} T^{3} - 5 p^{10} T^{4} - 3 p^{15} T^{5} + p^{21} T^{6} \) |
| good | 3 | \( 1 - 5102 T^{2} + 547829 p^{3} T^{4} - 397562596 p^{4} T^{6} + 547829 p^{17} T^{8} - 5102 p^{28} T^{10} + p^{42} T^{12} \) |
| 5 | \( 1 - 214718 T^{2} + 1269826431 p^{2} T^{4} - 4608734769796 p^{4} T^{6} + 1269826431 p^{16} T^{8} - 214718 p^{28} T^{10} + p^{42} T^{12} \) |
| 7 | \( ( 1 + 344 T + 1422245 T^{2} + 124668368 T^{3} + 1422245 p^{7} T^{4} + 344 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 11 | \( 1 - 64629022 T^{2} + 2363172524483783 T^{4} - \)\(55\!\cdots\!04\)\( T^{6} + 2363172524483783 p^{14} T^{8} - 64629022 p^{28} T^{10} + p^{42} T^{12} \) |
| 13 | \( 1 - 205410958 T^{2} + 1687860473809811 p T^{4} - \)\(16\!\cdots\!64\)\( T^{6} + 1687860473809811 p^{15} T^{8} - 205410958 p^{28} T^{10} + p^{42} T^{12} \) |
| 17 | \( ( 1 - 726 T + 323301023 T^{2} - 9708009717300 T^{3} + 323301023 p^{7} T^{4} - 726 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 19 | \( 1 - 2002416334 T^{2} + 2872909317854295863 T^{4} - \)\(28\!\cdots\!20\)\( T^{6} + 2872909317854295863 p^{14} T^{8} - 2002416334 p^{28} T^{10} + p^{42} T^{12} \) |
| 23 | \( ( 1 + 648 T + 9708521717 T^{2} + 6547475963760 T^{3} + 9708521717 p^{7} T^{4} + 648 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 29 | \( 1 - 47836636078 T^{2} + \)\(14\!\cdots\!79\)\( T^{4} - \)\(30\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!79\)\( p^{14} T^{8} - 47836636078 p^{28} T^{10} + p^{42} T^{12} \) |
| 31 | \( ( 1 + 1440 p T + 48354349725 T^{2} + 4324137771289408 T^{3} + 48354349725 p^{7} T^{4} + 1440 p^{15} T^{5} + p^{21} T^{6} )^{2} \) |
| 37 | \( 1 - 78937168126 T^{2} + \)\(11\!\cdots\!51\)\( T^{4} - \)\(18\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!51\)\( p^{14} T^{8} - 78937168126 p^{28} T^{10} + p^{42} T^{12} \) |
| 41 | \( ( 1 - 260622 T + 351195126263 T^{2} - 83834535574878564 T^{3} + 351195126263 p^{7} T^{4} - 260622 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 43 | \( 1 - 1505999929054 T^{2} + \)\(97\!\cdots\!71\)\( T^{4} - \)\(34\!\cdots\!32\)\( T^{6} + \)\(97\!\cdots\!71\)\( p^{14} T^{8} - 1505999929054 p^{28} T^{10} + p^{42} T^{12} \) |
| 47 | \( ( 1 - 783216 T + 1470820452333 T^{2} - 778339576797138720 T^{3} + 1470820452333 p^{7} T^{4} - 783216 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 53 | \( 1 - 3131635055902 T^{2} + \)\(64\!\cdots\!63\)\( T^{4} - \)\(86\!\cdots\!56\)\( T^{6} + \)\(64\!\cdots\!63\)\( p^{14} T^{8} - 3131635055902 p^{28} T^{10} + p^{42} T^{12} \) |
| 59 | \( 1 - 8312323804862 T^{2} + \)\(38\!\cdots\!03\)\( T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + \)\(38\!\cdots\!03\)\( p^{14} T^{8} - 8312323804862 p^{28} T^{10} + p^{42} T^{12} \) |
| 61 | \( 1 - 14350504934382 T^{2} + \)\(96\!\cdots\!03\)\( T^{4} - \)\(38\!\cdots\!24\)\( T^{6} + \)\(96\!\cdots\!03\)\( p^{14} T^{8} - 14350504934382 p^{28} T^{10} + p^{42} T^{12} \) |
| 67 | \( 1 - 35072892237678 T^{2} + \)\(51\!\cdots\!43\)\( T^{4} - \)\(41\!\cdots\!64\)\( T^{6} + \)\(51\!\cdots\!43\)\( p^{14} T^{8} - 35072892237678 p^{28} T^{10} + p^{42} T^{12} \) |
| 71 | \( ( 1 + 3798552 T + 17820666583269 T^{2} + 72937737977373055056 T^{3} + 17820666583269 p^{7} T^{4} + 3798552 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 73 | \( ( 1 - 1044782 T + 23207418111255 T^{2} - 22868192285089705636 T^{3} + 23207418111255 p^{7} T^{4} - 1044782 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 79 | \( ( 1 - 8007952 T + 49828330384013 T^{2} - \)\(25\!\cdots\!96\)\( T^{3} + 49828330384013 p^{7} T^{4} - 8007952 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 83 | \( 1 - 124932236904014 T^{2} + \)\(70\!\cdots\!51\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{6} + \)\(70\!\cdots\!51\)\( p^{14} T^{8} - 124932236904014 p^{28} T^{10} + p^{42} T^{12} \) |
| 89 | \( ( 1 - 1084542 T + 130289056617383 T^{2} - 94766843887733093316 T^{3} + 130289056617383 p^{7} T^{4} - 1084542 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 97 | \( ( 1 + 544154 T + 184934786492783 T^{2} + 86271378317799707180 T^{3} + 184934786492783 p^{7} T^{4} + 544154 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.12670376542152872730644551742, −11.20552536188170874799064326826, −11.11857004830857833873560851977, −10.87022029681820199795643416411, −10.44905461275987625856122430124, −10.27073147815272286823832202448, −9.773344448429515629654896213894, −9.674809299339302609644667991477, −9.200110809473735845947755289768, −8.929977715563553279398151659820, −8.181337482655758868344948114718, −8.052360778405481240336541638938, −7.38335003345176690415669664645, −7.15828558850434337911417394281, −6.92155637682498439363514104231, −6.40173411349288401009554966219, −6.14138368689667803508768059210, −5.39503189426215931113478803453, −4.76379405802049778035755527966, −4.53756424877732083482689976982, −3.90542560971273365534469377252, −3.26523589355173736921457070848, −2.59142600077458717024068467088, −1.51085202710998069083916310670, −0.933573956135023573850337573008,
0.933573956135023573850337573008, 1.51085202710998069083916310670, 2.59142600077458717024068467088, 3.26523589355173736921457070848, 3.90542560971273365534469377252, 4.53756424877732083482689976982, 4.76379405802049778035755527966, 5.39503189426215931113478803453, 6.14138368689667803508768059210, 6.40173411349288401009554966219, 6.92155637682498439363514104231, 7.15828558850434337911417394281, 7.38335003345176690415669664645, 8.052360778405481240336541638938, 8.181337482655758868344948114718, 8.929977715563553279398151659820, 9.200110809473735845947755289768, 9.674809299339302609644667991477, 9.773344448429515629654896213894, 10.27073147815272286823832202448, 10.44905461275987625856122430124, 10.87022029681820199795643416411, 11.11857004830857833873560851977, 11.20552536188170874799064326826, 12.12670376542152872730644551742
Plot not available for L-functions of degree greater than 10.
