| L(s) = 1 | + 2-s + 10·4-s − 15·5-s + 25·7-s + 33·8-s − 15·10-s + 58·11-s + 47·13-s + 25·14-s + 115·16-s − 68·17-s − 10·19-s − 150·20-s + 58·22-s − 51·23-s + 75·25-s + 47·26-s + 250·28-s + 350·29-s − 638·31-s + 414·32-s − 68·34-s − 375·35-s − 828·37-s − 10·38-s − 495·40-s + 179·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 5/4·4-s − 1.34·5-s + 1.34·7-s + 1.45·8-s − 0.474·10-s + 1.58·11-s + 1.00·13-s + 0.477·14-s + 1.79·16-s − 0.970·17-s − 0.120·19-s − 1.67·20-s + 0.562·22-s − 0.462·23-s + 3/5·25-s + 0.354·26-s + 1.68·28-s + 2.24·29-s − 3.69·31-s + 2.28·32-s − 0.342·34-s − 1.81·35-s − 3.67·37-s − 0.0426·38-s − 1.95·40-s + 0.681·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.267279878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267279878\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| good | 2 | \( 1 - T - 9 T^{2} - 7 p T^{3} + 11 p T^{4} + 29 p^{2} T^{5} + 41 p^{2} T^{6} + 29 p^{5} T^{7} + 11 p^{7} T^{8} - 7 p^{10} T^{9} - 9 p^{12} T^{10} - p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 25 T + 244 T^{2} - 7505 T^{3} - 10772 T^{4} + 3208115 T^{5} - 41593534 T^{6} + 3208115 p^{3} T^{7} - 10772 p^{6} T^{8} - 7505 p^{9} T^{9} + 244 p^{12} T^{10} - 25 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 58 T + 282 T^{2} + 136036 T^{3} - 3732386 T^{4} - 111996538 T^{5} + 10887063350 T^{6} - 111996538 p^{3} T^{7} - 3732386 p^{6} T^{8} + 136036 p^{9} T^{9} + 282 p^{12} T^{10} - 58 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 - 47 T + 3050 T^{2} - 288141 T^{3} + 10255156 T^{4} - 538326055 T^{5} + 35834260864 T^{6} - 538326055 p^{3} T^{7} + 10255156 p^{6} T^{8} - 288141 p^{9} T^{9} + 3050 p^{12} T^{10} - 47 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 + 2 p T + 12031 T^{2} + 14300 p T^{3} + 12031 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 + 5 T + 9800 T^{2} - 231055 T^{3} + 9800 p^{3} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 + 51 T - 18660 T^{2} + 684555 T^{3} + 183153756 T^{4} - 14586187377 T^{5} - 2242635291686 T^{6} - 14586187377 p^{3} T^{7} + 183153756 p^{6} T^{8} + 684555 p^{9} T^{9} - 18660 p^{12} T^{10} + 51 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 - 350 T + 69396 T^{2} - 6917320 T^{3} - 517207640 T^{4} + 388581844690 T^{5} - 78587536986586 T^{6} + 388581844690 p^{3} T^{7} - 517207640 p^{6} T^{8} - 6917320 p^{9} T^{9} + 69396 p^{12} T^{10} - 350 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 + 638 T + 182350 T^{2} + 1556452 p T^{3} + 13455116242 T^{4} + 2709597794894 T^{5} + 448785095846702 T^{6} + 2709597794894 p^{3} T^{7} + 13455116242 p^{6} T^{8} + 1556452 p^{10} T^{9} + 182350 p^{12} T^{10} + 638 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 + 414 T + 167991 T^{2} + 41362924 T^{3} + 167991 p^{3} T^{4} + 414 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 - 179 T - 22905 T^{2} + 3912848 T^{3} - 1998332639 T^{4} + 127749646003 T^{5} + 341742070644686 T^{6} + 127749646003 p^{3} T^{7} - 1998332639 p^{6} T^{8} + 3912848 p^{9} T^{9} - 22905 p^{12} T^{10} - 179 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 - 836 T + 239543 T^{2} - 80761716 T^{3} + 47721809782 T^{4} - 13871341463140 T^{5} + 2802082589374435 T^{6} - 13871341463140 p^{3} T^{7} + 47721809782 p^{6} T^{8} - 80761716 p^{9} T^{9} + 239543 p^{12} T^{10} - 836 p^{15} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 - 5 p T - 186564 T^{2} + 22762453 T^{3} + 24007674184 T^{4} - 355714744651 T^{5} - 2843354974813918 T^{6} - 355714744651 p^{3} T^{7} + 24007674184 p^{6} T^{8} + 22762453 p^{9} T^{9} - 186564 p^{12} T^{10} - 5 p^{16} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 + 505 T + 481759 T^{2} + 148865086 T^{3} + 481759 p^{3} T^{4} + 505 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 - 535 T - 78663 T^{2} + 288931294 T^{3} - 71306235053 T^{4} - 30710030725351 T^{5} + 36507468196017446 T^{6} - 30710030725351 p^{3} T^{7} - 71306235053 p^{6} T^{8} + 288931294 p^{9} T^{9} - 78663 p^{12} T^{10} - 535 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 104 T - 375715 T^{2} + 7139208 T^{3} + 59209422586 T^{4} + 3741915462248 T^{5} - 10521092248447379 T^{6} + 3741915462248 p^{3} T^{7} + 59209422586 p^{6} T^{8} + 7139208 p^{9} T^{9} - 375715 p^{12} T^{10} - 104 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 - 40 T - 525509 T^{2} - 4394696 T^{3} + 118975220434 T^{4} + 4665819395912 T^{5} - 29227192656155485 T^{6} + 4665819395912 p^{3} T^{7} + 118975220434 p^{6} T^{8} - 4394696 p^{9} T^{9} - 525509 p^{12} T^{10} - 40 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 452 T + 808810 T^{2} + 207368090 T^{3} + 808810 p^{3} T^{4} + 452 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 + 710 T + 1311287 T^{2} + 561111668 T^{3} + 1311287 p^{3} T^{4} + 710 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 - 634 T - 956873 T^{2} + 398955430 T^{3} + 776896953838 T^{4} - 160983685831738 T^{5} - 371177040015899773 T^{6} - 160983685831738 p^{3} T^{7} + 776896953838 p^{6} T^{8} + 398955430 p^{9} T^{9} - 956873 p^{12} T^{10} - 634 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 - 1734 T + 1009443 T^{2} + 57904194 T^{3} - 205697747874 T^{4} - 122842325158422 T^{5} + 209635494483596263 T^{6} - 122842325158422 p^{3} T^{7} - 205697747874 p^{6} T^{8} + 57904194 p^{9} T^{9} + 1009443 p^{12} T^{10} - 1734 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 - 852 T + 2335632 T^{2} - 1219193610 T^{3} + 2335632 p^{3} T^{4} - 852 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 1162851 T^{2} - 1406550016 T^{3} + 290919737478 T^{4} + 817804046327808 T^{5} + 780920935719579225 T^{6} + 817804046327808 p^{3} T^{7} + 290919737478 p^{6} T^{8} - 1406550016 p^{9} T^{9} - 1162851 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−5.52911121798089160820267471142, −5.37512536041597489741245992512, −5.33704660931981702498790704968, −5.26336592818974192484330983298, −4.75200273175611130245320914290, −4.56083443496268524887982704652, −4.44106294886030842788277060920, −4.27926158868649879108515208909, −4.27922892338312816076940371245, −3.88115627934379598153354938566, −3.63933704864360864537066731710, −3.55731821870574611346683560993, −3.43158062442353425444694886287, −3.42564324526607843813127527420, −2.69442909306504281969623421334, −2.64101080853975867462113051617, −2.38022650239251822366217767131, −2.22243307669330381926379856626, −1.63142408273439866584626582874, −1.62662391572516285998085203965, −1.57057442950052193918192380665, −1.29472192606745831488182616079, −0.890137360295800280281725224499, −0.65705203914212283968849517849, −0.078929629374126512802630378324,
0.078929629374126512802630378324, 0.65705203914212283968849517849, 0.890137360295800280281725224499, 1.29472192606745831488182616079, 1.57057442950052193918192380665, 1.62662391572516285998085203965, 1.63142408273439866584626582874, 2.22243307669330381926379856626, 2.38022650239251822366217767131, 2.64101080853975867462113051617, 2.69442909306504281969623421334, 3.42564324526607843813127527420, 3.43158062442353425444694886287, 3.55731821870574611346683560993, 3.63933704864360864537066731710, 3.88115627934379598153354938566, 4.27922892338312816076940371245, 4.27926158868649879108515208909, 4.44106294886030842788277060920, 4.56083443496268524887982704652, 4.75200273175611130245320914290, 5.26336592818974192484330983298, 5.33704660931981702498790704968, 5.37512536041597489741245992512, 5.52911121798089160820267471142
Plot not available for L-functions of degree greater than 10.
