| L(s) = 1 | + 162·4-s + 448·5-s + 3.23e4·11-s + 1.36e4·16-s + 7.25e4·20-s − 6.83e5·23-s − 1.14e6·25-s + 9.42e5·31-s − 3.80e6·37-s + 5.23e6·44-s − 1.58e7·47-s + 1.22e7·49-s + 3.54e7·53-s + 1.44e7·55-s + 2.96e7·59-s − 2.13e7·64-s + 3.94e7·67-s − 3.21e6·71-s + 6.10e6·80-s − 3.87e7·89-s − 1.10e8·92-s − 2.22e8·97-s − 1.85e8·100-s − 1.10e6·103-s + 1.63e8·113-s − 3.06e8·115-s + 3.68e8·121-s + ⋯ |
| L(s) = 1 | + 0.632·4-s + 0.716·5-s + 2.20·11-s + 0.207·16-s + 0.453·20-s − 2.44·23-s − 2.92·25-s + 1.02·31-s − 2.03·37-s + 1.39·44-s − 3.24·47-s + 2.12·49-s + 4.49·53-s + 1.58·55-s + 2.44·59-s − 1.27·64-s + 1.95·67-s − 0.126·71-s + 0.148·80-s − 0.618·89-s − 1.54·92-s − 2.50·97-s − 1.85·100-s − 0.00982·103-s + 1.00·113-s − 1.74·115-s + 1.72·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+4)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(12.41121064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.41121064\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 2938 p T + 5584439 p^{2} T^{2} - 59337484 p^{5} T^{3} + 5584439 p^{10} T^{4} - 2938 p^{17} T^{5} + p^{24} T^{6} \) |
| good | 2 | \( 1 - 81 p T^{2} + 789 p^{4} T^{4} + 21031 p^{10} T^{6} + 789 p^{20} T^{8} - 81 p^{33} T^{10} + p^{48} T^{12} \) |
| 5 | \( ( 1 - 224 T + 25856 p^{2} T^{2} - 8279534 p^{2} T^{3} + 25856 p^{10} T^{4} - 224 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 7 | \( 1 - 1751586 p T^{2} + 251817687273 p^{3} T^{4} - 25556340000207068 p^{5} T^{6} + 251817687273 p^{19} T^{8} - 1751586 p^{33} T^{10} + p^{48} T^{12} \) |
| 13 | \( 1 - 3254163342 T^{2} + 4990433215567563759 T^{4} - \)\(48\!\cdots\!56\)\( T^{6} + 4990433215567563759 p^{16} T^{8} - 3254163342 p^{32} T^{10} + p^{48} T^{12} \) |
| 17 | \( 1 - 36250223982 T^{2} + \)\(58\!\cdots\!79\)\( T^{4} - \)\(52\!\cdots\!36\)\( T^{6} + \)\(58\!\cdots\!79\)\( p^{16} T^{8} - 36250223982 p^{32} T^{10} + p^{48} T^{12} \) |
| 19 | \( 1 - 56326229646 T^{2} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(29\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!15\)\( p^{16} T^{8} - 56326229646 p^{32} T^{10} + p^{48} T^{12} \) |
| 23 | \( ( 1 + 341542 T + 158532994238 T^{2} + 26999761472338954 T^{3} + 158532994238 p^{8} T^{4} + 341542 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 29 | \( 1 - 1412429912166 T^{2} + \)\(92\!\cdots\!15\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(92\!\cdots\!15\)\( p^{16} T^{8} - 1412429912166 p^{32} T^{10} + p^{48} T^{12} \) |
| 31 | \( ( 1 - 471342 T + 2302578228774 T^{2} - 684878296100313386 T^{3} + 2302578228774 p^{8} T^{4} - 471342 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 37 | \( ( 1 + 1902408 T + 11067827666328 T^{2} + 13238766096402516286 T^{3} + 11067827666328 p^{8} T^{4} + 1902408 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 41 | \( 1 - 34135604017326 T^{2} + \)\(56\!\cdots\!15\)\( T^{4} - \)\(55\!\cdots\!20\)\( T^{6} + \)\(56\!\cdots\!15\)\( p^{16} T^{8} - 34135604017326 p^{32} T^{10} + p^{48} T^{12} \) |
| 43 | \( 1 - 16993443357702 T^{2} + \)\(10\!\cdots\!39\)\( T^{4} - \)\(93\!\cdots\!76\)\( T^{6} + \)\(10\!\cdots\!39\)\( p^{16} T^{8} - 16993443357702 p^{32} T^{10} + p^{48} T^{12} \) |
| 47 | \( ( 1 + 7914322 T + 64778997518303 T^{2} + \)\(28\!\cdots\!84\)\( T^{3} + 64778997518303 p^{8} T^{4} + 7914322 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 53 | \( ( 1 - 17738978 T + 290794219925903 T^{2} - \)\(24\!\cdots\!16\)\( T^{3} + 290794219925903 p^{8} T^{4} - 17738978 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 59 | \( ( 1 - 14807402 T + 386332407900254 T^{2} - \)\(38\!\cdots\!66\)\( T^{3} + 386332407900254 p^{8} T^{4} - 14807402 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 61 | \( 1 - 298340664593286 T^{2} + \)\(87\!\cdots\!15\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(87\!\cdots\!15\)\( p^{16} T^{8} - 298340664593286 p^{32} T^{10} + p^{48} T^{12} \) |
| 67 | \( ( 1 - 19709742 T + 873928607113398 T^{2} - \)\(98\!\cdots\!94\)\( T^{3} + 873928607113398 p^{8} T^{4} - 19709742 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 71 | \( ( 1 + 1609606 T + 1694098654931870 T^{2} + \)\(10\!\cdots\!90\)\( T^{3} + 1694098654931870 p^{8} T^{4} + 1609606 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 73 | \( 1 - 1855809765574062 T^{2} + \)\(13\!\cdots\!79\)\( T^{4} - \)\(72\!\cdots\!16\)\( T^{6} + \)\(13\!\cdots\!79\)\( p^{16} T^{8} - 1855809765574062 p^{32} T^{10} + p^{48} T^{12} \) |
| 79 | \( 1 - 4539802127333766 T^{2} + \)\(12\!\cdots\!15\)\( T^{4} - \)\(22\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!15\)\( p^{16} T^{8} - 4539802127333766 p^{32} T^{10} + p^{48} T^{12} \) |
| 83 | \( 1 - 6691963680142182 T^{2} + \)\(27\!\cdots\!79\)\( T^{4} - \)\(71\!\cdots\!36\)\( T^{6} + \)\(27\!\cdots\!79\)\( p^{16} T^{8} - 6691963680142182 p^{32} T^{10} + p^{48} T^{12} \) |
| 89 | \( ( 1 + 19392832 T + 5297816242291904 T^{2} + \)\(21\!\cdots\!86\)\( T^{3} + 5297816242291904 p^{8} T^{4} + 19392832 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 97 | \( ( 1 + 111092808 T + 19543603571024568 T^{2} + \)\(15\!\cdots\!26\)\( T^{3} + 19543603571024568 p^{8} T^{4} + 111092808 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
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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
−6.18559413834791036014447031898, −5.77935345543696829891306812546, −5.65938985159986547461199045753, −5.52489241110234082991843558547, −5.40983373179849900998185537778, −5.32114651511740927078072723390, −4.59097425849322133791512545347, −4.50102246356379774366585017404, −4.15266095005898331905876674942, −3.94754104095747548365492579575, −3.91173651893287300067195221566, −3.84482958520617002567729155594, −3.35534955878012455542826888072, −3.21478796037658261159979614970, −2.71050161091133845198043980041, −2.58645810552764790313435345801, −2.15488730441829709179034345693, −1.96679416569224031248531932558, −1.76080489384544735113173241391, −1.71502653057883163241973697329, −1.54196730492310359403635963661, −0.917956235821705968884873063862, −0.64079492478201354787348080808, −0.57637197194755649359544696041, −0.27039938394037972388821843780,
0.27039938394037972388821843780, 0.57637197194755649359544696041, 0.64079492478201354787348080808, 0.917956235821705968884873063862, 1.54196730492310359403635963661, 1.71502653057883163241973697329, 1.76080489384544735113173241391, 1.96679416569224031248531932558, 2.15488730441829709179034345693, 2.58645810552764790313435345801, 2.71050161091133845198043980041, 3.21478796037658261159979614970, 3.35534955878012455542826888072, 3.84482958520617002567729155594, 3.91173651893287300067195221566, 3.94754104095747548365492579575, 4.15266095005898331905876674942, 4.50102246356379774366585017404, 4.59097425849322133791512545347, 5.32114651511740927078072723390, 5.40983373179849900998185537778, 5.52489241110234082991843558547, 5.65938985159986547461199045753, 5.77935345543696829891306812546, 6.18559413834791036014447031898
Plot not available for L-functions of degree greater than 10.
