L(s) = 1 | + 2·2-s − 2·4-s + 2·5-s − 4·7-s − 8·8-s + 4·10-s + 18·11-s − 2·13-s − 8·14-s − 12·16-s + 4·17-s + 30·19-s − 4·20-s + 36·22-s − 60·23-s + 2·25-s − 4·26-s + 8·28-s + 18·29-s − 8·32-s + 8·34-s − 8·35-s + 46·37-s + 60·38-s − 16·40-s − 114·43-s − 36·44-s + ⋯ |
L(s) = 1 | + 2-s − 1/2·4-s + 2/5·5-s − 4/7·7-s − 8-s + 2/5·10-s + 1.63·11-s − 0.153·13-s − 4/7·14-s − 3/4·16-s + 4/17·17-s + 1.57·19-s − 1/5·20-s + 1.63·22-s − 2.60·23-s + 2/25·25-s − 0.153·26-s + 2/7·28-s + 0.620·29-s − 1/4·32-s + 4/17·34-s − 0.228·35-s + 1.24·37-s + 1.57·38-s − 2/5·40-s − 2.65·43-s − 0.818·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2331476347\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2331476347\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T + 3 p T^{2} - p^{3} T^{3} + 3 p^{3} T^{4} - p^{5} T^{5} + p^{6} T^{6} \) |
| 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
−6.86803044647321976267550908896, −6.71323850524147887331247068379, −6.71101019760727298429006340565, −6.28539900866339501356000722513, −6.13436494669926282911731377279, −6.11792080615838706809396421847, −5.90397729822676768941396341174, −5.50769028799827750079837184032, −5.35018006929628166230397902045, −5.17941745152808114205905029326, −4.76433458850358004087322060750, −4.62682744009068492294023668480, −4.43433460181694015089851240804, −4.23827692263010062386885872374, −4.18636931533633895672704997664, −3.61006054594482242283538638929, −3.41495026903471020384876688347, −3.21422849125596634704803606306, −3.04003759172283238292412603476, −2.87431317607089940919457236983, −2.09122136111130729778017329646, −1.83788685373977806222383602296, −1.51866660887037949973623794296, −1.16923206964364391420462402911, −0.096850873842387502485454755769,
0.096850873842387502485454755769, 1.16923206964364391420462402911, 1.51866660887037949973623794296, 1.83788685373977806222383602296, 2.09122136111130729778017329646, 2.87431317607089940919457236983, 3.04003759172283238292412603476, 3.21422849125596634704803606306, 3.41495026903471020384876688347, 3.61006054594482242283538638929, 4.18636931533633895672704997664, 4.23827692263010062386885872374, 4.43433460181694015089851240804, 4.62682744009068492294023668480, 4.76433458850358004087322060750, 5.17941745152808114205905029326, 5.35018006929628166230397902045, 5.50769028799827750079837184032, 5.90397729822676768941396341174, 6.11792080615838706809396421847, 6.13436494669926282911731377279, 6.28539900866339501356000722513, 6.71101019760727298429006340565, 6.71323850524147887331247068379, 6.86803044647321976267550908896
Plot not available for L-functions of degree greater than 10.