| L(s) = 1 | − 3·5-s − 6·7-s + 3·11-s + 6·13-s − 6·17-s + 3·19-s + 3·23-s + 3·25-s + 12·27-s + 24·29-s + 12·31-s + 18·35-s − 9·37-s + 18·41-s − 24·43-s − 15·47-s + 12·49-s − 9·53-s − 9·55-s + 24·59-s + 6·61-s − 18·65-s + 6·67-s − 18·77-s − 18·79-s + 60·83-s + 18·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 2.26·7-s + 0.904·11-s + 1.66·13-s − 1.45·17-s + 0.688·19-s + 0.625·23-s + 3/5·25-s + 2.30·27-s + 4.45·29-s + 2.15·31-s + 3.04·35-s − 1.47·37-s + 2.81·41-s − 3.65·43-s − 2.18·47-s + 12/7·49-s − 1.23·53-s − 1.21·55-s + 3.12·59-s + 0.768·61-s − 2.23·65-s + 0.733·67-s − 2.05·77-s − 2.02·79-s + 6.58·83-s + 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.348002476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.348002476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} )^{3} \) |
| 7 | \( 1 + 6 T + 24 T^{2} + 80 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( ( 1 - 2 p T^{3} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 3 T + 17 T^{3} - 84 T^{4} + 3 p T^{5} + 106 p T^{6} + 3 p^{2} T^{7} - 84 p^{2} T^{8} + 17 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 - 3 T + 15 T^{2} - 10 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3} \) |
| 19 | \( 1 - 3 T - 24 T^{2} + 161 T^{3} + 72 T^{4} - 1611 T^{5} + 6678 T^{6} - 1611 p T^{7} + 72 p^{2} T^{8} + 161 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 3 T - 45 T^{2} + 128 T^{3} + 1239 T^{4} - 1965 T^{5} - 26470 T^{6} - 1965 p T^{7} + 1239 p^{2} T^{8} + 128 p^{3} T^{9} - 45 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 12 T + 108 T^{2} - 670 T^{3} + 108 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 12 T + 39 T^{2} - 28 T^{3} + 378 T^{4} + 1908 T^{5} - 41121 T^{6} + 1908 p T^{7} + 378 p^{2} T^{8} - 28 p^{3} T^{9} + 39 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 9 T - 30 T^{2} - 141 T^{3} + 3084 T^{4} + 45 p T^{5} - 140560 T^{6} + 45 p^{2} T^{7} + 3084 p^{2} T^{8} - 141 p^{3} T^{9} - 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 9 T + 78 T^{2} - 357 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 12 T + 168 T^{2} + 1054 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 15 T + 180 T^{2} + 1031 T^{3} + 2580 T^{4} - 38325 T^{5} - 352870 T^{6} - 38325 p T^{7} + 2580 p^{2} T^{8} + 1031 p^{3} T^{9} + 180 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 9 T - 6 T^{2} + 123 T^{3} - 12 T^{4} - 29007 T^{5} - 222032 T^{6} - 29007 p T^{7} - 12 p^{2} T^{8} + 123 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{3} \) |
| 61 | \( 1 - 6 T - 60 T^{2} - 200 T^{3} + 2232 T^{4} + 29898 T^{5} - 206826 T^{6} + 29898 p T^{7} + 2232 p^{2} T^{8} - 200 p^{3} T^{9} - 60 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 6 T - 96 T^{2} + 832 T^{3} + 3708 T^{4} - 27990 T^{5} - 68142 T^{6} - 27990 p T^{7} + 3708 p^{2} T^{8} + 832 p^{3} T^{9} - 96 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + p T^{2} )^{6} \) |
| 73 | \( 1 - 111 T^{2} + 672 T^{3} + 4218 T^{4} - 37296 T^{5} - 8503 T^{6} - 37296 p T^{7} + 4218 p^{2} T^{8} + 672 p^{3} T^{9} - 111 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 + 18 T + 87 T^{2} + 114 T^{3} + 78 T^{4} - 882 p T^{5} - 1163581 T^{6} - 882 p^{2} T^{7} + 78 p^{2} T^{8} + 114 p^{3} T^{9} + 87 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 30 T + 540 T^{2} - 5884 T^{3} + 540 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 240 T^{2} + 84 T^{3} + 36240 T^{4} - 10080 T^{5} - 3714698 T^{6} - 10080 p T^{7} + 36240 p^{2} T^{8} + 84 p^{3} T^{9} - 240 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
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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
−6.01451699625565853755512659875, −5.71121927928558312000667255726, −5.24733633884423471292872188085, −5.10759435377205219947727066844, −5.07486358305256984886810395388, −4.88360650739755187963096683063, −4.67296260121069726689518210064, −4.61985150353475794569801361096, −4.33818931546539960639822824199, −4.25763233255412065159024056702, −3.77385364016134474081111336543, −3.72138069716077132040828023955, −3.52210124424857656561903401631, −3.50899894985610989971206169336, −3.17490205068513814529315321161, −3.11061385561824574953639568262, −2.80916059609410135212675562979, −2.62426234572303243918759667088, −2.49401664589686469415010324551, −2.17477049737854135177959879139, −1.65505289616137049195789820074, −1.32140756762208420421863705226, −0.891292974589946222813103796440, −0.884312786195528102555246972219, −0.45938387455822089526707781624,
0.45938387455822089526707781624, 0.884312786195528102555246972219, 0.891292974589946222813103796440, 1.32140756762208420421863705226, 1.65505289616137049195789820074, 2.17477049737854135177959879139, 2.49401664589686469415010324551, 2.62426234572303243918759667088, 2.80916059609410135212675562979, 3.11061385561824574953639568262, 3.17490205068513814529315321161, 3.50899894985610989971206169336, 3.52210124424857656561903401631, 3.72138069716077132040828023955, 3.77385364016134474081111336543, 4.25763233255412065159024056702, 4.33818931546539960639822824199, 4.61985150353475794569801361096, 4.67296260121069726689518210064, 4.88360650739755187963096683063, 5.07486358305256984886810395388, 5.10759435377205219947727066844, 5.24733633884423471292872188085, 5.71121927928558312000667255726, 6.01451699625565853755512659875
Plot not available for L-functions of degree greater than 10.
