| L(s) = 1 | − 14·2-s − 36·3-s − 196·4-s + 504·6-s + 1.38e3·8-s − 1.08e4·9-s + 4.69e4·11-s + 7.05e3·12-s + 5.29e4·16-s − 2.01e5·17-s + 1.52e5·18-s − 9.52e4·19-s − 6.57e5·22-s − 4.98e4·24-s + 9.47e5·25-s + 6.25e5·27-s + 5.45e5·32-s − 1.68e6·33-s + 2.81e6·34-s + 2.12e6·36-s + 1.33e6·38-s + 3.81e6·41-s − 9.88e6·43-s − 9.20e6·44-s − 1.90e6·48-s + 1.09e7·49-s − 1.32e7·50-s + ⋯ |
| L(s) = 1 | − 7/8·2-s − 4/9·3-s − 0.765·4-s + 7/18·6-s + 0.337·8-s − 1.65·9-s + 3.20·11-s + 0.340·12-s + 0.808·16-s − 2.40·17-s + 1.44·18-s − 0.731·19-s − 2.80·22-s − 0.150·24-s + 2.42·25-s + 1.17·27-s + 0.520·32-s − 1.42·33-s + 2.10·34-s + 1.26·36-s + 0.639·38-s + 1.35·41-s − 2.89·43-s − 2.45·44-s − 0.359·48-s + 1.90·49-s − 2.12·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+4)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.9489245169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9489245169\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 7 p T + 49 p^{3} T^{2} + 107 p^{6} T^{3} + 49 p^{11} T^{4} + 7 p^{17} T^{5} + p^{24} T^{6} \) |
| good | 3 | \( ( 1 + 2 p^{2} T + 1973 p T^{2} - 172 p^{3} T^{3} + 1973 p^{9} T^{4} + 2 p^{18} T^{5} + p^{24} T^{6} )^{2} \) |
| 5 | \( 1 - 189582 p T^{2} + 130696828227 p T^{4} - 2406033601733668 p^{3} T^{6} + 130696828227 p^{17} T^{8} - 189582 p^{33} T^{10} + p^{48} T^{12} \) |
| 7 | \( 1 - 1566666 p T^{2} + 150953794953 p^{3} T^{4} - 8689525475354828 p^{5} T^{6} + 150953794953 p^{19} T^{8} - 1566666 p^{33} T^{10} + p^{48} T^{12} \) |
| 11 | \( ( 1 - 23470 T + 781620575 T^{2} - 10217271642532 T^{3} + 781620575 p^{8} T^{4} - 23470 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 13 | \( 1 - 1977500742 T^{2} + 2369856232736193999 T^{4} - \)\(22\!\cdots\!96\)\( T^{6} + 2369856232736193999 p^{16} T^{8} - 1977500742 p^{32} T^{10} + p^{48} T^{12} \) |
| 17 | \( ( 1 + 5914 p T + 1307634847 p T^{2} + 1389520652768876 T^{3} + 1307634847 p^{9} T^{4} + 5914 p^{17} T^{5} + p^{24} T^{6} )^{2} \) |
| 19 | \( ( 1 + 47634 T + 10858923423 T^{2} - 1855478096374820 T^{3} + 10858923423 p^{8} T^{4} + 47634 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 23 | \( 1 - 130151035782 T^{2} + \)\(10\!\cdots\!19\)\( T^{4} - \)\(85\!\cdots\!56\)\( T^{6} + \)\(10\!\cdots\!19\)\( p^{16} T^{8} - 130151035782 p^{32} T^{10} + p^{48} T^{12} \) |
| 29 | \( 1 - 2162003976006 T^{2} + \)\(23\!\cdots\!35\)\( T^{4} - \)\(14\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!35\)\( p^{16} T^{8} - 2162003976006 p^{32} T^{10} + p^{48} T^{12} \) |
| 31 | \( 1 - 2084774115846 T^{2} + \)\(28\!\cdots\!15\)\( T^{4} - \)\(29\!\cdots\!20\)\( T^{6} + \)\(28\!\cdots\!15\)\( p^{16} T^{8} - 2084774115846 p^{32} T^{10} + p^{48} T^{12} \) |
| 37 | \( 1 - 14520610950342 T^{2} + \)\(10\!\cdots\!39\)\( T^{4} - \)\(45\!\cdots\!76\)\( T^{6} + \)\(10\!\cdots\!39\)\( p^{16} T^{8} - 14520610950342 p^{32} T^{10} + p^{48} T^{12} \) |
| 41 | \( ( 1 - 46550 p T + 11526892076495 T^{2} - 3479613964254273172 T^{3} + 11526892076495 p^{8} T^{4} - 46550 p^{17} T^{5} + p^{24} T^{6} )^{2} \) |
| 43 | \( ( 1 + 4940754 T + 42412589527647 T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + 42412589527647 p^{8} T^{4} + 4940754 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 47 | \( 1 - 107790912919302 T^{2} + \)\(52\!\cdots\!19\)\( T^{4} - \)\(15\!\cdots\!56\)\( T^{6} + \)\(52\!\cdots\!19\)\( p^{16} T^{8} - 107790912919302 p^{32} T^{10} + p^{48} T^{12} \) |
| 53 | \( 1 - 278586886481862 T^{2} + \)\(69\!\cdots\!03\)\( p T^{4} - \)\(29\!\cdots\!16\)\( T^{6} + \)\(69\!\cdots\!03\)\( p^{17} T^{8} - 278586886481862 p^{32} T^{10} + p^{48} T^{12} \) |
| 59 | \( ( 1 - 12621742 T + 462146058904799 T^{2} - \)\(36\!\cdots\!16\)\( T^{3} + 462146058904799 p^{8} T^{4} - 12621742 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 61 | \( 1 - 770402857907526 T^{2} + \)\(30\!\cdots\!35\)\( T^{4} - \)\(72\!\cdots\!80\)\( T^{6} + \)\(30\!\cdots\!35\)\( p^{16} T^{8} - 770402857907526 p^{32} T^{10} + p^{48} T^{12} \) |
| 67 | \( ( 1 - 23925102 T + 1197718266867999 T^{2} - \)\(18\!\cdots\!04\)\( T^{3} + 1197718266867999 p^{8} T^{4} - 23925102 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 71 | \( 1 - 2345316640410246 T^{2} + \)\(30\!\cdots\!95\)\( T^{4} - \)\(23\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!95\)\( p^{16} T^{8} - 2345316640410246 p^{32} T^{10} + p^{48} T^{12} \) |
| 73 | \( ( 1 + 27742458 T + 2200215907437519 T^{2} + \)\(43\!\cdots\!76\)\( T^{3} + 2200215907437519 p^{8} T^{4} + 27742458 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 79 | \( 1 - 3999742013149446 T^{2} + \)\(10\!\cdots\!95\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!95\)\( p^{16} T^{8} - 3999742013149446 p^{32} T^{10} + p^{48} T^{12} \) |
| 83 | \( ( 1 - 72323182 T + 7200759669989279 T^{2} - \)\(30\!\cdots\!84\)\( T^{3} + 7200759669989279 p^{8} T^{4} - 72323182 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 89 | \( ( 1 - 71086726 T + 11067563798758223 T^{2} - \)\(51\!\cdots\!40\)\( T^{3} + 11067563798758223 p^{8} T^{4} - 71086726 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 97 | \( ( 1 - 62596806 T + 19951560461187087 T^{2} - \)\(93\!\cdots\!72\)\( T^{3} + 19951560461187087 p^{8} T^{4} - 62596806 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
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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
−11.41050559091473875128009369784, −11.08442444126816421987667549340, −10.81293170545009752527416087717, −10.38623821652239110093721126597, −10.13664760427474661979708822558, −9.462171079760866224509810400717, −9.222949039176602476542289456717, −8.926535771024267960231693782593, −8.881597365220037595090290418049, −8.704841607960142089045314361872, −8.325569820342703437698541996951, −7.88141535453203042747199581836, −6.86630891488351354956893217016, −6.74532287802302545679632512458, −6.50997534143485794024088673381, −6.27636384248009048569426239994, −5.47223550074351082054634624986, −5.10256517488239345667532055047, −4.32943478763832320418513877186, −4.19527039942656978523606366812, −3.45707135943833539369736782408, −2.72305523666347011526242858192, −1.86957416678639384148861842939, −0.69075430298155837973348603044, −0.60588496463765312285620181653,
0.60588496463765312285620181653, 0.69075430298155837973348603044, 1.86957416678639384148861842939, 2.72305523666347011526242858192, 3.45707135943833539369736782408, 4.19527039942656978523606366812, 4.32943478763832320418513877186, 5.10256517488239345667532055047, 5.47223550074351082054634624986, 6.27636384248009048569426239994, 6.50997534143485794024088673381, 6.74532287802302545679632512458, 6.86630891488351354956893217016, 7.88141535453203042747199581836, 8.325569820342703437698541996951, 8.704841607960142089045314361872, 8.881597365220037595090290418049, 8.926535771024267960231693782593, 9.222949039176602476542289456717, 9.462171079760866224509810400717, 10.13664760427474661979708822558, 10.38623821652239110093721126597, 10.81293170545009752527416087717, 11.08442444126816421987667549340, 11.41050559091473875128009369784
Plot not available for L-functions of degree greater than 10.
