| L(s) = 1 | + 6·2-s + 13·4-s − 2·7-s + 6·8-s − 12·14-s − 22·16-s − 24·23-s + 15·25-s − 26·28-s − 30·29-s − 24·32-s − 4·37-s − 10·43-s − 144·46-s + 5·49-s + 90·50-s − 12·56-s − 180·58-s + 63·64-s + 12·67-s − 24·74-s − 6·79-s − 60·86-s − 312·92-s + 30·98-s + 195·100-s + 68·109-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 13/2·4-s − 0.755·7-s + 2.12·8-s − 3.20·14-s − 5.5·16-s − 5.00·23-s + 3·25-s − 4.91·28-s − 5.57·29-s − 4.24·32-s − 0.657·37-s − 1.52·43-s − 21.2·46-s + 5/7·49-s + 12.7·50-s − 1.60·56-s − 23.6·58-s + 63/8·64-s + 1.46·67-s − 2.78·74-s − 0.675·79-s − 6.46·86-s − 32.5·92-s + 3.03·98-s + 39/2·100-s + 6.51·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.380733974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.380733974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 2 T - T^{2} - 30 T^{3} - 50 T^{4} + 58 T^{5} + 715 T^{6} + 58 p T^{7} - 50 p^{2} T^{8} - 30 p^{3} T^{9} - p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| good | 2 | \( ( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{3} T^{5} + 31 T^{6} - 3 p^{4} T^{7} + 17 p^{2} T^{8} - 3 p^{5} T^{9} + 7 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 5 | \( 1 - 3 p T^{2} + 84 T^{4} - 457 T^{6} + 3921 T^{8} - 20514 T^{10} + 80841 T^{12} - 20514 p^{2} T^{14} + 3921 p^{4} T^{16} - 457 p^{6} T^{18} + 84 p^{8} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \) |
| 11 | \( ( 1 + 2 p T^{2} + 2 p^{2} T^{4} - 6 p T^{5} + 2665 T^{6} - 6 p^{2} T^{7} + 2 p^{4} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( 1 + 33 T^{2} + 27 p T^{4} + 2144 T^{6} + 53457 T^{8} + 906351 T^{10} + 9826878 T^{12} + 906351 p^{2} T^{14} + 53457 p^{4} T^{16} + 2144 p^{6} T^{18} + 27 p^{9} T^{20} + 33 p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( ( 1 + 48 T^{2} + 1338 T^{4} + 26845 T^{6} + 1338 p^{2} T^{8} + 48 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 51 T^{2} + 1599 T^{4} - 35471 T^{6} + 1599 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 + 12 T + 130 T^{2} + 984 T^{3} + 7082 T^{4} + 39654 T^{5} + 209833 T^{6} + 39654 p T^{7} + 7082 p^{2} T^{8} + 984 p^{3} T^{9} + 130 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 15 T + 184 T^{2} + 1635 T^{3} + 13205 T^{4} + 85158 T^{5} + 500899 T^{6} + 85158 p T^{7} + 13205 p^{2} T^{8} + 1635 p^{3} T^{9} + 184 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( 1 + 57 T^{2} + 3216 T^{4} + 97787 T^{6} + 2702697 T^{8} + 47442966 T^{10} + 1072931313 T^{12} + 47442966 p^{2} T^{14} + 2702697 p^{4} T^{16} + 97787 p^{6} T^{18} + 3216 p^{8} T^{20} + 57 p^{10} T^{22} + p^{12} T^{24} \) |
| 37 | \( ( 1 + T + 107 T^{2} + 73 T^{3} + 107 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( 1 - 174 T^{2} + 15930 T^{4} - 1039210 T^{6} + 54933612 T^{8} - 2550462528 T^{10} + 108692576859 T^{12} - 2550462528 p^{2} T^{14} + 54933612 p^{4} T^{16} - 1039210 p^{6} T^{18} + 15930 p^{8} T^{20} - 174 p^{10} T^{22} + p^{12} T^{24} \) |
| 43 | \( ( 1 + 5 T - 58 T^{2} - 51 T^{3} + 2155 T^{4} - 7106 T^{5} - 129149 T^{6} - 7106 p T^{7} + 2155 p^{2} T^{8} - 51 p^{3} T^{9} - 58 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( 1 - 189 T^{2} + 18726 T^{4} - 1271869 T^{6} + 67940493 T^{8} - 3218131254 T^{10} + 150306249081 T^{12} - 3218131254 p^{2} T^{14} + 67940493 p^{4} T^{16} - 1271869 p^{6} T^{18} + 18726 p^{8} T^{20} - 189 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( ( 1 - 200 T^{2} + 20924 T^{4} - 1371431 T^{6} + 20924 p^{2} T^{8} - 200 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( 1 - 135 T^{2} + 4578 T^{4} + 47801 T^{6} + 11150547 T^{8} - 1706849076 T^{10} + 108381578457 T^{12} - 1706849076 p^{2} T^{14} + 11150547 p^{4} T^{16} + 47801 p^{6} T^{18} + 4578 p^{8} T^{20} - 135 p^{10} T^{22} + p^{12} T^{24} \) |
| 61 | \( 1 + 342 T^{2} + 66840 T^{4} + 9095042 T^{6} + 949804386 T^{8} + 78844563978 T^{10} + 5324903072391 T^{12} + 78844563978 p^{2} T^{14} + 949804386 p^{4} T^{16} + 9095042 p^{6} T^{18} + 66840 p^{8} T^{20} + 342 p^{10} T^{22} + p^{12} T^{24} \) |
| 67 | \( ( 1 - 6 T - 150 T^{2} + 506 T^{3} + 17268 T^{4} - 28236 T^{5} - 1220289 T^{6} - 28236 p T^{7} + 17268 p^{2} T^{8} + 506 p^{3} T^{9} - 150 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 263 T^{2} + 31517 T^{4} - 2539307 T^{6} + 31517 p^{2} T^{8} - 263 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 159 T^{2} + 10563 T^{4} - 623423 T^{6} + 10563 p^{2} T^{8} - 159 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 3 T - 204 T^{2} - 151 T^{3} + 27357 T^{4} + 24 p T^{5} - 31623 p T^{6} + 24 p^{2} T^{7} + 27357 p^{2} T^{8} - 151 p^{3} T^{9} - 204 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 - 264 T^{2} + 27306 T^{4} - 2832334 T^{6} + 401548758 T^{8} - 35501027934 T^{10} + 2433955301391 T^{12} - 35501027934 p^{2} T^{14} + 401548758 p^{4} T^{16} - 2832334 p^{6} T^{18} + 27306 p^{8} T^{20} - 264 p^{10} T^{22} + p^{12} T^{24} \) |
| 89 | \( ( 1 + 210 T^{2} + 31902 T^{4} + 3230233 T^{6} + 31902 p^{2} T^{8} + 210 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( 1 + 513 T^{2} + 147894 T^{4} + 30072485 T^{6} + 4748892315 T^{8} + 607745530428 T^{10} + 64510991774769 T^{12} + 607745530428 p^{2} T^{14} + 4748892315 p^{4} T^{16} + 30072485 p^{6} T^{18} + 147894 p^{8} T^{20} + 513 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.49587522070343263889297810480, −4.41875839478143503784048271524, −4.13927899129749112506449740701, −3.95678900458276712235893860172, −3.92614880930788595164598737937, −3.88902011780119719378323057258, −3.87575815086337407193920184575, −3.69752586245828272573291308311, −3.64138506773329048308771201641, −3.48679074714212651949736984927, −3.42815671689783544475660044488, −3.27288882154243200825932417482, −3.09815308548138871674193506266, −3.04335907719612786680512741462, −2.83564868048947459662711409844, −2.70897374221943998806039585883, −2.34770525027868703003146903112, −2.17910684348331092153140832136, −2.08174355276751018544557511780, −1.90316335246636950398364869551, −1.89149283792605881726851118836, −1.84994846296907642085155337872, −1.08014189312548087383128657237, −1.03846462802600001614286503433, −0.22890934771112124956133219779,
0.22890934771112124956133219779, 1.03846462802600001614286503433, 1.08014189312548087383128657237, 1.84994846296907642085155337872, 1.89149283792605881726851118836, 1.90316335246636950398364869551, 2.08174355276751018544557511780, 2.17910684348331092153140832136, 2.34770525027868703003146903112, 2.70897374221943998806039585883, 2.83564868048947459662711409844, 3.04335907719612786680512741462, 3.09815308548138871674193506266, 3.27288882154243200825932417482, 3.42815671689783544475660044488, 3.48679074714212651949736984927, 3.64138506773329048308771201641, 3.69752586245828272573291308311, 3.87575815086337407193920184575, 3.88902011780119719378323057258, 3.92614880930788595164598737937, 3.95678900458276712235893860172, 4.13927899129749112506449740701, 4.41875839478143503784048271524, 4.49587522070343263889297810480
Plot not available for L-functions of degree greater than 10.
