| L(s) = 1 | + 3·5-s − 3·7-s + 6·8-s + 6·11-s − 3·13-s − 6·17-s − 6·23-s + 9·25-s + 3·29-s − 6·31-s − 9·35-s + 30·37-s + 18·40-s − 12·41-s − 12·43-s + 3·49-s − 24·53-s + 18·55-s − 18·56-s − 18·59-s + 3·61-s + 11·64-s − 9·65-s − 6·67-s + 18·73-s − 18·77-s − 6·79-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s + 2.12·8-s + 1.80·11-s − 0.832·13-s − 1.45·17-s − 1.25·23-s + 9/5·25-s + 0.557·29-s − 1.07·31-s − 1.52·35-s + 4.93·37-s + 2.84·40-s − 1.87·41-s − 1.82·43-s + 3/7·49-s − 3.29·53-s + 2.42·55-s − 2.40·56-s − 2.34·59-s + 0.384·61-s + 11/8·64-s − 1.11·65-s − 0.733·67-s + 2.10·73-s − 2.05·77-s − 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050654069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050654069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 + T + T^{2} )^{3} \) |
| good | 2 | \( ( 1 - 3 T^{3} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 - 3 T + 9 T^{3} - 18 T^{4} + 33 T^{5} - 56 T^{6} + 33 p T^{7} - 18 p^{2} T^{8} + 9 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 6 T + 36 T^{3} + 144 T^{4} - 510 T^{5} - 74 T^{6} - 510 p T^{7} + 144 p^{2} T^{8} + 36 p^{3} T^{9} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 3 T + 6 T^{2} + 77 T^{3} + 72 T^{4} + 171 T^{5} + 3912 T^{6} + 171 p T^{7} + 72 p^{2} T^{8} + 77 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 3 T + 27 T^{2} + 114 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 + 9 T^{2} - 56 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 6 T - 9 T^{2} - 90 T^{3} - 90 T^{4} - 1146 T^{5} - 8885 T^{6} - 1146 p T^{7} - 90 p^{2} T^{8} - 90 p^{3} T^{9} - 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 3 T - 42 T^{2} + 267 T^{3} + 492 T^{4} - 4359 T^{5} + 9880 T^{6} - 4359 p T^{7} + 492 p^{2} T^{8} + 267 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3} \) |
| 37 | \( ( 1 - 5 T + p T^{2} )^{6} \) |
| 41 | \( 1 + 12 T + 21 T^{2} - 348 T^{3} - 858 T^{4} + 276 p T^{5} + 107233 T^{6} + 276 p^{2} T^{7} - 858 p^{2} T^{8} - 348 p^{3} T^{9} + 21 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 12 T + 60 T^{2} + 248 T^{3} - 1116 T^{4} - 32580 T^{5} - 266250 T^{6} - 32580 p T^{7} - 1116 p^{2} T^{8} + 248 p^{3} T^{9} + 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 117 T^{2} + 48 T^{3} + 8190 T^{4} - 2808 T^{5} - 435161 T^{6} - 2808 p T^{7} + 8190 p^{2} T^{8} + 48 p^{3} T^{9} - 117 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 12 T + 180 T^{2} + 1266 T^{3} + 180 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 18 T + 135 T^{2} + 258 T^{3} - 3726 T^{4} - 41670 T^{5} - 353441 T^{6} - 41670 p T^{7} - 3726 p^{2} T^{8} + 258 p^{3} T^{9} + 135 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 3 T - 96 T^{2} + 41 T^{3} + 4086 T^{4} + 8577 T^{5} - 220368 T^{6} + 8577 p T^{7} + 4086 p^{2} T^{8} + 41 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 6 T - 96 T^{2} - 292 T^{3} + 5328 T^{4} - 7650 T^{5} - 433722 T^{6} - 7650 p T^{7} + 5328 p^{2} T^{8} - 292 p^{3} T^{9} - 96 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 132 T^{2} + 108 T^{3} + 132 p T^{4} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 9 T + 207 T^{2} - 1298 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 6 T - 132 T^{2} - 364 T^{3} + 10836 T^{4} - 6570 T^{5} - 997698 T^{6} - 6570 p T^{7} + 10836 p^{2} T^{8} - 364 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 12 T - 105 T^{2} + 852 T^{3} + 18294 T^{4} - 780 p T^{5} - 1308125 T^{6} - 780 p^{2} T^{7} + 18294 p^{2} T^{8} + 852 p^{3} T^{9} - 105 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 15 T + 315 T^{2} + 2622 T^{3} + 315 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 12 T - 147 T^{2} - 796 T^{3} + 30726 T^{4} + 612 p T^{5} - 3146871 T^{6} + 612 p^{2} T^{7} + 30726 p^{2} T^{8} - 796 p^{3} T^{9} - 147 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.87646030596889412233419789543, −5.56786684044337190569677272556, −5.56057447616494571271081874261, −5.28597662720470994052125869723, −4.99692377705375041502021025481, −4.69146886866227810530422733347, −4.64465181446402820079628155614, −4.57332572702976668641586552203, −4.43369544346722806330130678334, −4.32528498169946433259902604946, −4.11673003216853946851522576400, −3.68174122814131003664506410270, −3.63064374525355337760403192760, −3.48070172015234276088371885998, −3.03880509645944097490488191831, −3.02344082744635252027399671844, −2.59418071420702314399250088866, −2.46048629532007490062993137915, −2.40899931425066165141704433777, −1.91384436702685900172554370375, −1.65155921756506443363791051498, −1.51980847526357409828092862039, −1.31046971586801636628892728414, −1.12052124353740472277859379283, −0.18072925506282542923039038145,
0.18072925506282542923039038145, 1.12052124353740472277859379283, 1.31046971586801636628892728414, 1.51980847526357409828092862039, 1.65155921756506443363791051498, 1.91384436702685900172554370375, 2.40899931425066165141704433777, 2.46048629532007490062993137915, 2.59418071420702314399250088866, 3.02344082744635252027399671844, 3.03880509645944097490488191831, 3.48070172015234276088371885998, 3.63064374525355337760403192760, 3.68174122814131003664506410270, 4.11673003216853946851522576400, 4.32528498169946433259902604946, 4.43369544346722806330130678334, 4.57332572702976668641586552203, 4.64465181446402820079628155614, 4.69146886866227810530422733347, 4.99692377705375041502021025481, 5.28597662720470994052125869723, 5.56057447616494571271081874261, 5.56786684044337190569677272556, 5.87646030596889412233419789543
Plot not available for L-functions of degree greater than 10.
