L(s) = 1 | + 2·3-s + 8·7-s + 2·9-s − 4·13-s + 12·19-s + 16·21-s − 2·27-s − 4·37-s − 8·39-s − 12·43-s − 20·49-s + 24·57-s + 12·61-s + 16·63-s − 28·67-s − 7·81-s − 32·91-s − 8·97-s + 56·103-s + 12·109-s − 8·111-s − 8·117-s + 127-s − 24·129-s + 131-s + 96·133-s + 137-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 3.02·7-s + 2/3·9-s − 1.10·13-s + 2.75·19-s + 3.49·21-s − 0.384·27-s − 0.657·37-s − 1.28·39-s − 1.82·43-s − 2.85·49-s + 3.17·57-s + 1.53·61-s + 2.01·63-s − 3.42·67-s − 7/9·81-s − 3.35·91-s − 0.812·97-s + 5.51·103-s + 1.14·109-s − 0.759·111-s − 0.739·117-s + 0.0887·127-s − 2.11·129-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.932155943\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.932155943\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T + 2 T^{2} + 2 T^{3} - 5 T^{4} + 20 T^{5} - 28 T^{6} + 20 p T^{7} - 5 p^{2} T^{8} + 2 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
good | 5 | \( 1 - 6 p T^{4} - 49 T^{8} + 12796 T^{12} - 49 p^{4} T^{16} - 6 p^{9} T^{20} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 2 T + 15 T^{2} - 20 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 11 | \( 1 - 62 T^{4} + 11023 T^{8} - 2631620 T^{12} + 11023 p^{4} T^{16} - 62 p^{8} T^{20} + p^{12} T^{24} \) |
| 13 | \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} - 25 T^{4} + 412 T^{5} + 892 T^{6} + 412 p T^{7} - 25 p^{2} T^{8} - 6 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 62 T^{2} + 1903 T^{4} - 38180 T^{6} + 1903 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 6 T + 18 T^{2} - 82 T^{3} + 539 T^{4} - 2636 T^{5} + 9476 T^{6} - 2636 p T^{7} + 539 p^{2} T^{8} - 82 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 86 T^{2} + 3791 T^{4} - 105684 T^{6} + 3791 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( 1 - 830 T^{4} + 2253679 T^{8} - 1165110596 T^{12} + 2253679 p^{4} T^{16} - 830 p^{8} T^{20} + p^{12} T^{24} \) |
| 31 | \( ( 1 - 150 T^{2} + 10019 T^{4} - 392444 T^{6} + 10019 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} - 54 T^{3} + 567 T^{4} + 8764 T^{5} + 17852 T^{6} + 8764 p T^{7} + 567 p^{2} T^{8} - 54 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 + 138 T^{2} + 7887 T^{4} + 320492 T^{6} + 7887 p^{2} T^{8} + 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + 6 T + 18 T^{2} + 226 T^{3} + 4235 T^{4} + 18188 T^{5} + 58436 T^{6} + 18188 p T^{7} + 4235 p^{2} T^{8} + 226 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 170 T^{2} + 15791 T^{4} + 908172 T^{6} + 15791 p^{2} T^{8} + 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( 1 + 7714 T^{4} + 19237903 T^{8} + 30633057916 T^{12} + 19237903 p^{4} T^{16} + 7714 p^{8} T^{20} + p^{12} T^{24} \) |
| 59 | \( ( 1 - 30 T + 450 T^{2} - 3458 T^{3} + 3915 T^{4} + 231740 T^{5} - 2735068 T^{6} + 231740 p T^{7} + 3915 p^{2} T^{8} - 3458 p^{3} T^{9} + 450 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )( 1 + 30 T + 450 T^{2} + 3458 T^{3} + 3915 T^{4} - 231740 T^{5} - 2735068 T^{6} - 231740 p T^{7} + 3915 p^{2} T^{8} + 3458 p^{3} T^{9} + 450 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} ) \) |
| 61 | \( ( 1 - 6 T + 18 T^{2} - 430 T^{3} - 121 T^{4} + 30796 T^{5} - 90148 T^{6} + 30796 p T^{7} - 121 p^{2} T^{8} - 430 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 + 14 T + 98 T^{2} + 706 T^{3} + 9435 T^{4} + 112164 T^{5} + 894884 T^{6} + 112164 p T^{7} + 9435 p^{2} T^{8} + 706 p^{3} T^{9} + 98 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 230 T^{2} + 30127 T^{4} - 2527028 T^{6} + 30127 p^{2} T^{8} - 230 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 166 T^{2} + 19007 T^{4} - 1414164 T^{6} + 19007 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 358 T^{2} + 58915 T^{4} - 5817628 T^{6} + 58915 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 - 1374 T^{4} + 18563631 T^{8} - 336062521604 T^{12} + 18563631 p^{4} T^{16} - 1374 p^{8} T^{20} + p^{12} T^{24} \) |
| 89 | \( ( 1 + 322 T^{2} + 51919 T^{4} + 5548348 T^{6} + 51919 p^{2} T^{8} + 322 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 2 T + 163 T^{2} - 220 T^{3} + 163 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.50986112245900434132508594839, −4.21317630814752928008318784895, −4.13347330741189315537039763727, −4.02971811404217210353355398658, −3.91122648897124150984836282583, −3.85580128647303538734194880823, −3.77225188269583600126601590197, −3.49887565166340526902831036167, −3.47759715048646383968528783028, −3.21997576937661597155969495608, −3.12422475107545287240073771539, −3.05385202407856414320334319242, −3.03620387649918954668145143874, −2.78101865293312824691971321029, −2.54807376021836737486554806937, −2.53507531515238817052259123893, −2.36168515248227211072694865671, −2.05501230773373667726771941559, −1.95193416007845952453880695989, −1.68992746023688664867774879247, −1.57396781394862861913967863184, −1.44832875490441251134371734677, −1.34359846347367033329461073281, −1.33883260711732846891386188754, −0.44765714708931943290980177321,
0.44765714708931943290980177321, 1.33883260711732846891386188754, 1.34359846347367033329461073281, 1.44832875490441251134371734677, 1.57396781394862861913967863184, 1.68992746023688664867774879247, 1.95193416007845952453880695989, 2.05501230773373667726771941559, 2.36168515248227211072694865671, 2.53507531515238817052259123893, 2.54807376021836737486554806937, 2.78101865293312824691971321029, 3.03620387649918954668145143874, 3.05385202407856414320334319242, 3.12422475107545287240073771539, 3.21997576937661597155969495608, 3.47759715048646383968528783028, 3.49887565166340526902831036167, 3.77225188269583600126601590197, 3.85580128647303538734194880823, 3.91122648897124150984836282583, 4.02971811404217210353355398658, 4.13347330741189315537039763727, 4.21317630814752928008318784895, 4.50986112245900434132508594839
Plot not available for L-functions of degree greater than 10.