L(s) = 1 | + 3·2-s + 6·4-s + 3·5-s − 3·7-s + 9·8-s + 9·10-s + 6·11-s + 3·13-s − 9·14-s + 12·16-s − 12·17-s − 6·19-s + 18·20-s + 18·22-s + 12·23-s + 15·25-s + 9·26-s − 18·28-s + 9·29-s + 3·31-s + 12·32-s − 36·34-s − 9·35-s − 6·37-s − 18·38-s + 27·40-s + 3·43-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 3·4-s + 1.34·5-s − 1.13·7-s + 3.18·8-s + 2.84·10-s + 1.80·11-s + 0.832·13-s − 2.40·14-s + 3·16-s − 2.91·17-s − 1.37·19-s + 4.02·20-s + 3.83·22-s + 2.50·23-s + 3·25-s + 1.76·26-s − 3.40·28-s + 1.67·29-s + 0.538·31-s + 2.12·32-s − 6.17·34-s − 1.52·35-s − 0.986·37-s − 2.91·38-s + 4.26·40-s + 0.457·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.714661431\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.714661431\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + 3 T^{2} - 3 T^{4} + 3 p T^{5} - 11 T^{6} + 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 6 p T^{5} - 371 T^{6} - 6 p^{2} T^{7} + 69 p^{2} T^{8} + 9 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 6 T - 6 T^{2} + 18 T^{3} + 492 T^{4} - 852 T^{5} - 2873 T^{6} - 852 p T^{7} + 492 p^{2} T^{8} + 18 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 + 3 T + 51 T^{2} + 97 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 12 T + 48 T^{2} - 54 T^{3} + 420 T^{4} - 6060 T^{5} + 37591 T^{6} - 6060 p T^{7} + 420 p^{2} T^{8} - 54 p^{3} T^{9} + 48 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 9414 p T^{7} - 579 p^{2} T^{8} - 81 p^{3} T^{9} + 30 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 3 T - 6 T^{2} - 319 T^{3} + 171 T^{4} + 1962 T^{5} + 62727 T^{6} + 1962 p T^{7} + 171 p^{2} T^{8} - 319 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 1026 p T^{7} + 8322 p^{2} T^{8} + 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 3 T - 78 T^{2} + 405 T^{3} + 2481 T^{4} - 11064 T^{5} - 57089 T^{6} - 11064 p T^{7} + 2481 p^{2} T^{8} + 405 p^{3} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 6 T + 150 T^{2} + 639 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 3 T - 96 T^{2} - 495 T^{3} + 3615 T^{4} + 15798 T^{5} - 107021 T^{6} + 15798 p T^{7} + 3615 p^{2} T^{8} - 495 p^{3} T^{9} - 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 6 T - 132 T^{2} - 418 T^{3} + 13698 T^{4} + 19134 T^{5} - 893289 T^{6} + 19134 p T^{7} + 13698 p^{2} T^{8} - 418 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 50094 p T^{7} + 15318 p^{2} T^{8} + 518 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 303 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 195678 p T^{7} + 25767 p^{2} T^{8} - 499 p^{3} T^{9} + 84 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 126648 p T^{7} + 8088 p^{2} T^{8} - 702 p^{3} T^{9} + 30 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 12 T + 204 T^{2} + 1323 T^{3} + 204 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.72826094754002162447343403048, −6.61444128359880553100548282014, −6.58905225828936102131683508168, −6.40420101292222859163302143262, −6.18851063398547620558297262052, −5.97319602943244698246172172806, −5.89007315023916187007989011715, −5.68671443817530915704681156292, −5.36421495847388876699018500862, −5.08339352163665890410551723168, −4.73892170285543167822416058950, −4.70116061828204160052833276461, −4.48263794856912570888418305475, −4.37330280820422888416478285161, −4.15941807685087078981683397483, −3.90727407255354225909979301245, −3.36748418655730523022974973046, −3.15593454568155724252765376838, −3.15142591218481922883376292724, −2.88912242770314900545470895722, −2.50273273181646134525430891596, −2.35125114336309548659765692959, −1.83261589329923896951674637946, −1.44847883687483930650634110133, −1.18828745570387212178649372787,
1.18828745570387212178649372787, 1.44847883687483930650634110133, 1.83261589329923896951674637946, 2.35125114336309548659765692959, 2.50273273181646134525430891596, 2.88912242770314900545470895722, 3.15142591218481922883376292724, 3.15593454568155724252765376838, 3.36748418655730523022974973046, 3.90727407255354225909979301245, 4.15941807685087078981683397483, 4.37330280820422888416478285161, 4.48263794856912570888418305475, 4.70116061828204160052833276461, 4.73892170285543167822416058950, 5.08339352163665890410551723168, 5.36421495847388876699018500862, 5.68671443817530915704681156292, 5.89007315023916187007989011715, 5.97319602943244698246172172806, 6.18851063398547620558297262052, 6.40420101292222859163302143262, 6.58905225828936102131683508168, 6.61444128359880553100548282014, 6.72826094754002162447343403048
Plot not available for L-functions of degree greater than 10.