| L(s) = 1 | − 2·5-s + 4·7-s + 18·11-s + 2·13-s + 4·17-s + 30·19-s + 60·23-s + 2·25-s − 18·29-s − 8·35-s − 46·37-s − 114·43-s − 162·49-s + 78·53-s − 36·55-s − 206·59-s − 30·61-s − 4·65-s − 226·67-s − 260·71-s + 72·77-s − 318·83-s − 8·85-s + 8·91-s − 60·95-s − 4·97-s + 142·101-s + ⋯ |
| L(s) = 1 | − 2/5·5-s + 4/7·7-s + 1.63·11-s + 2/13·13-s + 4/17·17-s + 1.57·19-s + 2.60·23-s + 2/25·25-s − 0.620·29-s − 0.228·35-s − 1.24·37-s − 2.65·43-s − 3.30·49-s + 1.47·53-s − 0.654·55-s − 3.49·59-s − 0.491·61-s − 0.0615·65-s − 3.37·67-s − 3.66·71-s + 0.935·77-s − 3.83·83-s − 0.0941·85-s + 8/91·91-s − 0.631·95-s − 0.0412·97-s + 1.40·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5038881349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5038881349\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2 T + 2 T^{2} - 14 T^{3} - 369 T^{4} + 636 T^{5} + 2108 T^{6} + 636 p^{2} T^{7} - 369 p^{4} T^{8} - 14 p^{6} T^{9} + 2 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} \) |
| 7 | \( ( 1 - 2 T + 87 T^{2} - 332 T^{3} + 87 p^{2} T^{4} - 2 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( 1 - 18 T + 162 T^{2} - 2146 T^{3} + 17759 T^{4} - 65756 T^{5} + 609308 T^{6} - 65756 p^{2} T^{7} + 17759 p^{4} T^{8} - 2146 p^{6} T^{9} + 162 p^{8} T^{10} - 18 p^{10} T^{11} + p^{12} T^{12} \) |
| 13 | \( 1 - 2 T + 2 T^{2} - 1554 T^{3} - 7825 T^{4} + 453380 T^{5} + 316348 T^{6} + 453380 p^{2} T^{7} - 7825 p^{4} T^{8} - 1554 p^{6} T^{9} + 2 p^{8} T^{10} - 2 p^{10} T^{11} + p^{12} T^{12} \) |
| 17 | \( ( 1 - 2 T + 607 T^{2} + 388 T^{3} + 607 p^{2} T^{4} - 2 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( 1 - 30 T + 450 T^{2} - 12014 T^{3} + 441215 T^{4} - 8004292 T^{5} + 113750108 T^{6} - 8004292 p^{2} T^{7} + 441215 p^{4} T^{8} - 12014 p^{6} T^{9} + 450 p^{8} T^{10} - 30 p^{10} T^{11} + p^{12} T^{12} \) |
| 23 | \( ( 1 - 30 T + 1751 T^{2} - 30772 T^{3} + 1751 p^{2} T^{4} - 30 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 29 | \( 1 + 18 T + 162 T^{2} - 4894 T^{3} + 124463 T^{4} + 24625372 T^{5} + 435069308 T^{6} + 24625372 p^{2} T^{7} + 124463 p^{4} T^{8} - 4894 p^{6} T^{9} + 162 p^{8} T^{10} + 18 p^{10} T^{11} + p^{12} T^{12} \) |
| 31 | \( 1 - 3846 T^{2} + 7131791 T^{4} - 8361808916 T^{6} + 7131791 p^{4} T^{8} - 3846 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 + 46 T + 1058 T^{2} - 6594 T^{3} - 356337 T^{4} + 78343460 T^{5} + 4002544124 T^{6} + 78343460 p^{2} T^{7} - 356337 p^{4} T^{8} - 6594 p^{6} T^{9} + 1058 p^{8} T^{10} + 46 p^{10} T^{11} + p^{12} T^{12} \) |
| 41 | \( 1 - 5094 T^{2} + 15050223 T^{4} - 31243096276 T^{6} + 15050223 p^{4} T^{8} - 5094 p^{8} T^{10} + p^{12} T^{12} \) |
| 43 | \( 1 + 114 T + 6498 T^{2} + 241730 T^{3} + 12357983 T^{4} + 838941724 T^{5} + 44553879452 T^{6} + 838941724 p^{2} T^{7} + 12357983 p^{4} T^{8} + 241730 p^{6} T^{9} + 6498 p^{8} T^{10} + 114 p^{10} T^{11} + p^{12} T^{12} \) |
| 47 | \( 1 - 4678 T^{2} + 12462287 T^{4} - 24905944212 T^{6} + 12462287 p^{4} T^{8} - 4678 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 78 T + 3042 T^{2} - 270110 T^{3} + 31648463 T^{4} - 1389102820 T^{5} + 48555101564 T^{6} - 1389102820 p^{2} T^{7} + 31648463 p^{4} T^{8} - 270110 p^{6} T^{9} + 3042 p^{8} T^{10} - 78 p^{10} T^{11} + p^{12} T^{12} \) |
| 59 | \( 1 + 206 T + 21218 T^{2} + 1942462 T^{3} + 171214239 T^{4} + 11916831972 T^{5} + 708622973852 T^{6} + 11916831972 p^{2} T^{7} + 171214239 p^{4} T^{8} + 1942462 p^{6} T^{9} + 21218 p^{8} T^{10} + 206 p^{10} T^{11} + p^{12} T^{12} \) |
| 61 | \( 1 + 30 T + 450 T^{2} + 111694 T^{3} + 33268655 T^{4} + 582006980 T^{5} + 8727089468 T^{6} + 582006980 p^{2} T^{7} + 33268655 p^{4} T^{8} + 111694 p^{6} T^{9} + 450 p^{8} T^{10} + 30 p^{10} T^{11} + p^{12} T^{12} \) |
| 67 | \( 1 + 226 T + 25538 T^{2} + 2083538 T^{3} + 120508479 T^{4} + 5203289532 T^{5} + 268963196252 T^{6} + 5203289532 p^{2} T^{7} + 120508479 p^{4} T^{8} + 2083538 p^{6} T^{9} + 25538 p^{8} T^{10} + 226 p^{10} T^{11} + p^{12} T^{12} \) |
| 71 | \( ( 1 + 130 T + 11575 T^{2} + 918796 T^{3} + 11575 p^{2} T^{4} + 130 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 73 | \( 1 - 13126 T^{2} + 103571951 T^{4} - 653716749588 T^{6} + 103571951 p^{4} T^{8} - 13126 p^{8} T^{10} + p^{12} T^{12} \) |
| 79 | \( 1 - 70 T^{2} + 84324175 T^{4} + 17226941804 T^{6} + 84324175 p^{4} T^{8} - 70 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( 1 + 318 T + 50562 T^{2} + 6712846 T^{3} + 819490815 T^{4} + 81203275140 T^{5} + 6918697616348 T^{6} + 81203275140 p^{2} T^{7} + 819490815 p^{4} T^{8} + 6712846 p^{6} T^{9} + 50562 p^{8} T^{10} + 318 p^{10} T^{11} + p^{12} T^{12} \) |
| 89 | \( 1 - 31238 T^{2} + 466178479 T^{4} - 4433595811988 T^{6} + 466178479 p^{4} T^{8} - 31238 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( ( 1 + 2 T + 10687 T^{2} + 557564 T^{3} + 10687 p^{2} T^{4} + 2 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−4.82635575658603751992454383208, −4.70452146020924675718165008964, −4.68385162325867274685260264000, −4.61959622782471950373483026047, −4.54447308331471076089620390771, −3.98269768072816514404493531606, −3.94927002705153903519685234139, −3.89970897589967938317425331019, −3.57838372346465538282961829297, −3.34872415953437987878980189435, −3.18314755479663810315066963096, −3.08262745336625497548507604340, −3.07826589304159698531944584307, −2.88654953472402596806598487418, −2.78720011658679223310657015287, −2.28119950917667611394813277304, −1.85335179355938475298648613854, −1.80046093408871849594241342659, −1.75224893082775659043646071319, −1.30971585975397005246660388110, −1.26281794773343853176502669899, −1.18909179932744731685652238985, −0.991273404440174421890630557136, −0.24586260173011279235043628546, −0.10489654131579401405016466770,
0.10489654131579401405016466770, 0.24586260173011279235043628546, 0.991273404440174421890630557136, 1.18909179932744731685652238985, 1.26281794773343853176502669899, 1.30971585975397005246660388110, 1.75224893082775659043646071319, 1.80046093408871849594241342659, 1.85335179355938475298648613854, 2.28119950917667611394813277304, 2.78720011658679223310657015287, 2.88654953472402596806598487418, 3.07826589304159698531944584307, 3.08262745336625497548507604340, 3.18314755479663810315066963096, 3.34872415953437987878980189435, 3.57838372346465538282961829297, 3.89970897589967938317425331019, 3.94927002705153903519685234139, 3.98269768072816514404493531606, 4.54447308331471076089620390771, 4.61959622782471950373483026047, 4.68385162325867274685260264000, 4.70452146020924675718165008964, 4.82635575658603751992454383208
Plot not available for L-functions of degree greater than 10.
