| L(s) = 1 | − 6·2-s − 9·3-s + 24·4-s + 54·6-s − 21·7-s − 80·8-s + 54·9-s + 38·11-s − 216·12-s − 42·13-s + 126·14-s + 240·16-s − 30·17-s − 324·18-s + 57·19-s + 189·21-s − 228·22-s − 36·23-s + 720·24-s − 175·25-s + 252·26-s − 270·27-s − 504·28-s + 76·29-s + 166·31-s − 672·32-s − 342·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 3.67·6-s − 1.13·7-s − 3.53·8-s + 2·9-s + 1.04·11-s − 5.19·12-s − 0.896·13-s + 2.40·14-s + 15/4·16-s − 0.428·17-s − 4.24·18-s + 0.688·19-s + 1.96·21-s − 2.20·22-s − 0.326·23-s + 6.12·24-s − 7/5·25-s + 1.90·26-s − 1.92·27-s − 3.40·28-s + 0.486·29-s + 0.961·31-s − 3.71·32-s − 1.80·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 7 p^{2} T^{2} + 1064 T^{3} + 7 p^{5} T^{4} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 38 T + 4129 T^{2} - 97924 T^{3} + 4129 p^{3} T^{4} - 38 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 42 T + 6379 T^{2} + 184604 T^{3} + 6379 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 30 T + 13695 T^{2} + 278972 T^{3} + 13695 p^{3} T^{4} + 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 36 T + 18565 T^{2} + 992376 T^{3} + 18565 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 76 T + 211 p T^{2} + 4807952 T^{3} + 211 p^{4} T^{4} - 76 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 166 T + 75293 T^{2} - 8775316 T^{3} + 75293 p^{3} T^{4} - 166 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 48 T + 117739 T^{2} - 6647664 T^{3} + 117739 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 102 T + 45591 T^{2} + 14596620 T^{3} + 45591 p^{3} T^{4} - 102 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 52 T + 223609 T^{2} + 7631544 T^{3} + 223609 p^{3} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 48 T + 199741 T^{2} - 12874872 T^{3} + 199741 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 224 T + 276623 T^{2} - 39780008 T^{3} + 276623 p^{3} T^{4} - 224 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1004 T + 834169 T^{2} - 397907912 T^{3} + 834169 p^{3} T^{4} - 1004 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 450 T + 290827 T^{2} + 191190604 T^{3} + 290827 p^{3} T^{4} + 450 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 906 T + 1023657 T^{2} + 504932924 T^{3} + 1023657 p^{3} T^{4} + 906 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 946 T + 923353 T^{2} - 546808796 T^{3} + 923353 p^{3} T^{4} - 946 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 862 T + 36983 T^{2} - 312720508 T^{3} + 36983 p^{3} T^{4} + 862 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 148 T + 149821 T^{2} + 393066216 T^{3} + 149821 p^{3} T^{4} - 148 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 778 T + 1885885 T^{2} + 900149116 T^{3} + 1885885 p^{3} T^{4} + 778 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 226 T + 952839 T^{2} + 627413308 T^{3} + 952839 p^{3} T^{4} + 226 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2200 T + 3657815 T^{2} + 3892880192 T^{3} + 3657815 p^{3} T^{4} + 2200 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.405694502414496120383874223338, −8.752046597058744714810555319711, −8.709678560462165098523147038106, −8.655318567592197616504934040307, −7.78680089985895265535755065310, −7.71067162075241610914365519508, −7.56287205090650517403840845193, −7.12689816361981885260474888891, −6.86381376076701580941876726947, −6.60489014416128675285336202800, −6.23151902622764996477252721893, −6.17882936044691037959103450225, −5.96228503377750605546962865940, −5.29224836207262156521762534106, −5.18075551362465971539090485533, −4.85283898169297330079317730664, −3.94358173359349410867211565202, −3.90271364470487325853467280366, −3.85104824517842829656910576259, −2.78758337654086005500480969957, −2.52075894700835529630104633740, −2.45932926458221738779920383413, −1.37659125665115121829996176608, −1.32915809657444210904725241169, −1.02446314920538743099175162483, 0, 0, 0,
1.02446314920538743099175162483, 1.32915809657444210904725241169, 1.37659125665115121829996176608, 2.45932926458221738779920383413, 2.52075894700835529630104633740, 2.78758337654086005500480969957, 3.85104824517842829656910576259, 3.90271364470487325853467280366, 3.94358173359349410867211565202, 4.85283898169297330079317730664, 5.18075551362465971539090485533, 5.29224836207262156521762534106, 5.96228503377750605546962865940, 6.17882936044691037959103450225, 6.23151902622764996477252721893, 6.60489014416128675285336202800, 6.86381376076701580941876726947, 7.12689816361981885260474888891, 7.56287205090650517403840845193, 7.71067162075241610914365519508, 7.78680089985895265535755065310, 8.655318567592197616504934040307, 8.709678560462165098523147038106, 8.752046597058744714810555319711, 9.405694502414496120383874223338
