| L(s) = 1 | + 2·5-s + 2·7-s + 13-s − 4·19-s − 14·23-s + 7·25-s + 4·29-s − 30·31-s + 4·35-s − 6·37-s − 4·41-s − 3·43-s + 18·47-s − 21·49-s − 6·53-s − 13·61-s + 2·65-s + 9·67-s + 18·71-s − 19·73-s + 11·79-s − 8·83-s − 16·89-s + 2·91-s − 8·95-s + 2·97-s + 22·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.277·13-s − 0.917·19-s − 2.91·23-s + 7/5·25-s + 0.742·29-s − 5.38·31-s + 0.676·35-s − 0.986·37-s − 0.624·41-s − 0.457·43-s + 2.62·47-s − 3·49-s − 0.824·53-s − 1.66·61-s + 0.248·65-s + 1.09·67-s + 2.13·71-s − 2.22·73-s + 1.23·79-s − 0.878·83-s − 1.69·89-s + 0.209·91-s − 0.820·95-s + 0.203·97-s + 2.18·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2668696946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2668696946\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4 T + 17 T^{2} + 136 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 2 T - 3 T^{2} + 2 T^{3} - 2 T^{4} + 34 T^{5} - 31 T^{6} + 34 p T^{7} - 2 p^{2} T^{8} + 2 p^{3} T^{9} - 3 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( ( 1 - T + 12 T^{2} - 17 T^{3} + 12 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 + 9 T^{2} - 36 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - T - 17 T^{2} + 40 T^{3} + 61 T^{4} - 223 T^{5} + 854 T^{6} - 223 p T^{7} + 61 p^{2} T^{8} + 40 p^{3} T^{9} - 17 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 23 | \( 1 + 14 T + 99 T^{2} + 382 T^{3} + 346 T^{4} - 7726 T^{5} - 57613 T^{6} - 7726 p T^{7} + 346 p^{2} T^{8} + 382 p^{3} T^{9} + 99 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 4 T - 39 T^{2} + 52 T^{3} + 886 T^{4} + 1916 T^{5} - 33907 T^{6} + 1916 p T^{7} + 886 p^{2} T^{8} + 52 p^{3} T^{9} - 39 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 15 T + 144 T^{2} + 29 p T^{3} + 144 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + T + p T^{2} )^{6} \) |
| 41 | \( 1 + 4 T - 75 T^{2} - 100 T^{3} + 3622 T^{4} - 2012 T^{5} - 174751 T^{6} - 2012 p T^{7} + 3622 p^{2} T^{8} - 100 p^{3} T^{9} - 75 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 3 T - 99 T^{2} - 218 T^{3} + 6207 T^{4} + 6951 T^{5} - 285738 T^{6} + 6951 p T^{7} + 6207 p^{2} T^{8} - 218 p^{3} T^{9} - 99 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 53 | \( 1 + 6 T - 51 T^{2} - 102 T^{3} + 1086 T^{4} - 11334 T^{5} - 107183 T^{6} - 11334 p T^{7} + 1086 p^{2} T^{8} - 102 p^{3} T^{9} - 51 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 153 T^{2} + 72 T^{3} + 14382 T^{4} - 5508 T^{5} - 969077 T^{6} - 5508 p T^{7} + 14382 p^{2} T^{8} + 72 p^{3} T^{9} - 153 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 + 13 T - 25 T^{2} - 504 T^{3} + 6133 T^{4} + 34211 T^{5} - 181514 T^{6} + 34211 p T^{7} + 6133 p^{2} T^{8} - 504 p^{3} T^{9} - 25 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 9 T - 39 T^{2} + 250 T^{3} + 375 T^{4} + 24519 T^{5} - 284658 T^{6} + 24519 p T^{7} + 375 p^{2} T^{8} + 250 p^{3} T^{9} - 39 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 18 T + 99 T^{2} - 234 T^{3} + 306 T^{4} + 55062 T^{5} - 871373 T^{6} + 55062 p T^{7} + 306 p^{2} T^{8} - 234 p^{3} T^{9} + 99 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 19 T + 59 T^{2} + 252 T^{3} + 17041 T^{4} + 91889 T^{5} - 279578 T^{6} + 91889 p T^{7} + 17041 p^{2} T^{8} + 252 p^{3} T^{9} + 59 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 11 T - 119 T^{2} + 494 T^{3} + 21403 T^{4} - 27611 T^{5} - 1863994 T^{6} - 27611 p T^{7} + 21403 p^{2} T^{8} + 494 p^{3} T^{9} - 119 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 4 T + 169 T^{2} + 472 T^{3} + 169 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 16 T - 87 T^{2} - 424 T^{3} + 39826 T^{4} + 163780 T^{5} - 2350663 T^{6} + 163780 p T^{7} + 39826 p^{2} T^{8} - 424 p^{3} T^{9} - 87 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 2 T - 19 T^{2} - 2166 T^{3} + 418 T^{4} + 14486 T^{5} + 2859997 T^{6} + 14486 p T^{7} + 418 p^{2} T^{8} - 2166 p^{3} T^{9} - 19 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.55392819542778961398097000770, −4.45202662315622975132039704083, −4.41821572399686948429587212679, −4.13996704940570783161401196729, −3.84870057796152887253456632305, −3.74526929174479374753780484246, −3.69518418115996276483139552655, −3.45290698426526208624439960836, −3.43716241866183184997895035283, −3.34070918428583191638113892583, −3.28754615230875397822875563933, −2.65951122310780824906612709034, −2.64267100416645858283394770764, −2.61666538947727443858617402679, −2.23247227691159418958044663409, −2.02448728082545304952933696029, −1.98475022074641266274034760751, −1.88451397674878543224435337605, −1.87622345109952783893745608490, −1.37783453169289343612390786265, −1.36224134516743826028820053926, −1.15393194193708779487454245128, −0.863744173324517766121023486179, −0.16713998856345718772603329228, −0.13348308783236207734273688825,
0.13348308783236207734273688825, 0.16713998856345718772603329228, 0.863744173324517766121023486179, 1.15393194193708779487454245128, 1.36224134516743826028820053926, 1.37783453169289343612390786265, 1.87622345109952783893745608490, 1.88451397674878543224435337605, 1.98475022074641266274034760751, 2.02448728082545304952933696029, 2.23247227691159418958044663409, 2.61666538947727443858617402679, 2.64267100416645858283394770764, 2.65951122310780824906612709034, 3.28754615230875397822875563933, 3.34070918428583191638113892583, 3.43716241866183184997895035283, 3.45290698426526208624439960836, 3.69518418115996276483139552655, 3.74526929174479374753780484246, 3.84870057796152887253456632305, 4.13996704940570783161401196729, 4.41821572399686948429587212679, 4.45202662315622975132039704083, 4.55392819542778961398097000770
Plot not available for L-functions of degree greater than 10.
