| L(s) = 1 | + 3·2-s + 4-s + 2·5-s + 26·7-s − 6·8-s + 6·10-s − 15·13-s + 78·14-s − 16-s + 10·17-s − 46·19-s + 2·20-s + 24·23-s + 47·25-s − 45·26-s + 26·28-s − 66·29-s + 27·32-s + 30·34-s + 52·35-s − 138·38-s − 12·40-s − 24·41-s + 11·43-s + 72·46-s + 26·47-s + 239·49-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 1/4·4-s + 2/5·5-s + 26/7·7-s − 3/4·8-s + 3/5·10-s − 1.15·13-s + 39/7·14-s − 0.0625·16-s + 0.588·17-s − 2.42·19-s + 1/10·20-s + 1.04·23-s + 1.87·25-s − 1.73·26-s + 0.928·28-s − 2.27·29-s + 0.843·32-s + 0.882·34-s + 1.48·35-s − 3.63·38-s − 0.299·40-s − 0.585·41-s + 0.255·43-s + 1.56·46-s + 0.553·47-s + 4.87·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.714286085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.714286085\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + 46 T + 1383 T^{2} + 1676 p T^{3} + 1383 p^{2} T^{4} + 46 p^{4} T^{5} + p^{6} T^{6} \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} - 15 T^{3} + 5 p^{2} T^{4} - 27 T^{5} - T^{6} - 27 p^{2} T^{7} + 5 p^{6} T^{8} - 15 p^{6} T^{9} + p^{11} T^{10} - 3 p^{10} T^{11} + p^{12} T^{12} \) |
| 5 | \( 1 - 2 T - 43 T^{2} + 18 p T^{3} + 34 p^{2} T^{4} - 874 T^{5} - 18011 T^{6} - 874 p^{2} T^{7} + 34 p^{6} T^{8} + 18 p^{7} T^{9} - 43 p^{8} T^{10} - 2 p^{10} T^{11} + p^{12} T^{12} \) |
| 7 | \( ( 1 - 13 T + 134 T^{2} - 841 T^{3} + 134 p^{2} T^{4} - 13 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( ( 1 + 9 p T^{2} + 304 T^{3} + 9 p^{3} T^{4} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 + 15 T + 335 T^{2} + 300 p T^{3} + 32765 T^{4} + 3465 p T^{5} + 734054 T^{6} + 3465 p^{3} T^{7} + 32765 p^{4} T^{8} + 300 p^{7} T^{9} + 335 p^{8} T^{10} + 15 p^{10} T^{11} + p^{12} T^{12} \) |
| 17 | \( 1 - 10 T - 395 T^{2} + 6466 T^{3} + 44890 T^{4} - 799570 T^{5} + 1692077 T^{6} - 799570 p^{2} T^{7} + 44890 p^{4} T^{8} + 6466 p^{6} T^{9} - 395 p^{8} T^{10} - 10 p^{10} T^{11} + p^{12} T^{12} \) |
| 23 | \( 1 - 24 T - 891 T^{2} + 13112 T^{3} + 796206 T^{4} - 4908840 T^{5} - 419866119 T^{6} - 4908840 p^{2} T^{7} + 796206 p^{4} T^{8} + 13112 p^{6} T^{9} - 891 p^{8} T^{10} - 24 p^{10} T^{11} + p^{12} T^{12} \) |
| 29 | \( 1 + 66 T + 4043 T^{2} + 171006 T^{3} + 6809138 T^{4} + 221239218 T^{5} + 6993379739 T^{6} + 221239218 p^{2} T^{7} + 6809138 p^{4} T^{8} + 171006 p^{6} T^{9} + 4043 p^{8} T^{10} + 66 p^{10} T^{11} + p^{12} T^{12} \) |
| 31 | \( 1 - 3733 T^{2} + 7198070 T^{4} - 8589921313 T^{6} + 7198070 p^{4} T^{8} - 3733 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 - 2301 T^{2} + 3048342 T^{4} - 3163158425 T^{6} + 3048342 p^{4} T^{8} - 2301 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( 1 + 24 T + 2771 T^{2} + 61896 T^{3} + 2356262 T^{4} - 54820104 T^{5} + 1146448727 T^{6} - 54820104 p^{2} T^{7} + 2356262 p^{4} T^{8} + 61896 p^{6} T^{9} + 2771 p^{8} T^{10} + 24 p^{10} T^{11} + p^{12} T^{12} \) |
| 43 | \( 1 - 11 T - 5437 T^{2} + 468 p T^{3} + 20167681 T^{4} - 37345585 T^{5} - 43238202698 T^{6} - 37345585 p^{2} T^{7} + 20167681 p^{4} T^{8} + 468 p^{7} T^{9} - 5437 p^{8} T^{10} - 11 p^{10} T^{11} + p^{12} T^{12} \) |
| 47 | \( 1 - 26 T - 907 T^{2} + 283794 T^{3} - 5215334 T^{4} - 202235170 T^{5} + 39122708077 T^{6} - 202235170 p^{2} T^{7} - 5215334 p^{4} T^{8} + 283794 p^{6} T^{9} - 907 p^{8} T^{10} - 26 p^{10} T^{11} + p^{12} T^{12} \) |
| 53 | \( 1 + 180 T + 19443 T^{2} + 1555740 T^{3} + 110933502 T^{4} + 7296213492 T^{5} + 419317225855 T^{6} + 7296213492 p^{2} T^{7} + 110933502 p^{4} T^{8} + 1555740 p^{6} T^{9} + 19443 p^{8} T^{10} + 180 p^{10} T^{11} + p^{12} T^{12} \) |
| 59 | \( 1 + 162 T + 17555 T^{2} + 1426734 T^{3} + 87243434 T^{4} + 4615710834 T^{5} + 265630952507 T^{6} + 4615710834 p^{2} T^{7} + 87243434 p^{4} T^{8} + 1426734 p^{6} T^{9} + 17555 p^{8} T^{10} + 162 p^{10} T^{11} + p^{12} T^{12} \) |
| 61 | \( 1 + 141 T + 12075 T^{2} + 497204 T^{3} - 10655151 T^{4} - 3883509825 T^{5} - 307926913914 T^{6} - 3883509825 p^{2} T^{7} - 10655151 p^{4} T^{8} + 497204 p^{6} T^{9} + 12075 p^{8} T^{10} + 141 p^{10} T^{11} + p^{12} T^{12} \) |
| 67 | \( 1 + 63 T + 12647 T^{2} + 713412 T^{3} + 87514253 T^{4} + 4184717949 T^{5} + 445391002646 T^{6} + 4184717949 p^{2} T^{7} + 87514253 p^{4} T^{8} + 713412 p^{6} T^{9} + 12647 p^{8} T^{10} + 63 p^{10} T^{11} + p^{12} T^{12} \) |
| 71 | \( 1 - 372 T + 75795 T^{2} - 11036124 T^{3} + 1263566262 T^{4} - 118063927428 T^{5} + 9166752875239 T^{6} - 118063927428 p^{2} T^{7} + 1263566262 p^{4} T^{8} - 11036124 p^{6} T^{9} + 75795 p^{8} T^{10} - 372 p^{10} T^{11} + p^{12} T^{12} \) |
| 73 | \( 1 - 103 T + 1219 T^{2} + 302980 T^{3} - 27279935 T^{4} + 1419724787 T^{5} - 65157351466 T^{6} + 1419724787 p^{2} T^{7} - 27279935 p^{4} T^{8} + 302980 p^{6} T^{9} + 1219 p^{8} T^{10} - 103 p^{10} T^{11} + p^{12} T^{12} \) |
| 79 | \( 1 + 123 T + 23399 T^{2} + 2257788 T^{3} + 281744093 T^{4} + 275766615 p T^{5} + 339528854 p^{2} T^{6} + 275766615 p^{3} T^{7} + 281744093 p^{4} T^{8} + 2257788 p^{6} T^{9} + 23399 p^{8} T^{10} + 123 p^{10} T^{11} + p^{12} T^{12} \) |
| 83 | \( ( 1 - 126 T + 25575 T^{2} - 1793732 T^{3} + 25575 p^{2} T^{4} - 126 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( 1 + 642 T + 206859 T^{2} + 44600382 T^{3} + 7147480218 T^{4} + 891909981330 T^{5} + 88632420639379 T^{6} + 891909981330 p^{2} T^{7} + 7147480218 p^{4} T^{8} + 44600382 p^{6} T^{9} + 206859 p^{8} T^{10} + 642 p^{10} T^{11} + p^{12} T^{12} \) |
| 97 | \( 1 + 12 T + 899 T^{2} + 10212 T^{3} - 18381130 T^{4} - 3423508740 T^{5} + 902696454071 T^{6} - 3423508740 p^{2} T^{7} - 18381130 p^{4} T^{8} + 10212 p^{6} T^{9} + 899 p^{8} T^{10} + 12 p^{10} T^{11} + p^{12} T^{12} \) |
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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
−6.82957440701504592960880104934, −6.51828991937071483147868387635, −6.43212382070534489431598265091, −6.02921277364822162061382557859, −5.97112151800625340654194573376, −5.70368436180890617545069629804, −5.39027309367532600657350148772, −5.15269330571445192874602284120, −5.13355208499972933254737860788, −4.80904916845132666853380906035, −4.80738533942226284698767250046, −4.67333057434990471743608274193, −4.37825163417346417460519279224, −4.28969071020278770271956840654, −3.99083230508453840355131077720, −3.64079735509608469227193632118, −3.31807540632195250449342338073, −2.96067154828541059784462023168, −2.94174367412872288221104885647, −2.20947897427584202234681928214, −2.20572473742520175652590216244, −1.62612383278772591596437753402, −1.57157107967013314014852328612, −1.33810331491247913736206653133, −0.36986254549519978206089716352,
0.36986254549519978206089716352, 1.33810331491247913736206653133, 1.57157107967013314014852328612, 1.62612383278772591596437753402, 2.20572473742520175652590216244, 2.20947897427584202234681928214, 2.94174367412872288221104885647, 2.96067154828541059784462023168, 3.31807540632195250449342338073, 3.64079735509608469227193632118, 3.99083230508453840355131077720, 4.28969071020278770271956840654, 4.37825163417346417460519279224, 4.67333057434990471743608274193, 4.80738533942226284698767250046, 4.80904916845132666853380906035, 5.13355208499972933254737860788, 5.15269330571445192874602284120, 5.39027309367532600657350148772, 5.70368436180890617545069629804, 5.97112151800625340654194573376, 6.02921277364822162061382557859, 6.43212382070534489431598265091, 6.51828991937071483147868387635, 6.82957440701504592960880104934
Plot not available for L-functions of degree greater than 10.
