| L(s) = 1 | + 3·3-s − 3·5-s − 3·7-s + 6·9-s − 2·11-s + 5·13-s − 9·15-s − 4·17-s − 14·19-s − 9·21-s + 3·23-s + 25-s + 10·27-s − 4·29-s − 6·31-s − 6·33-s + 9·35-s − 6·37-s + 15·39-s + 10·41-s − 21·43-s − 18·45-s − 2·47-s + 6·49-s − 12·51-s − 13·53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s − 1.13·7-s + 2·9-s − 0.603·11-s + 1.38·13-s − 2.32·15-s − 0.970·17-s − 3.21·19-s − 1.96·21-s + 0.625·23-s + 1/5·25-s + 1.92·27-s − 0.742·29-s − 1.07·31-s − 1.04·33-s + 1.52·35-s − 0.986·37-s + 2.40·39-s + 1.56·41-s − 3.20·43-s − 2.68·45-s − 0.291·47-s + 6/7·49-s − 1.68·51-s − 1.78·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 3 T + 8 T^{2} + 2 p T^{3} + 8 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 24 T^{2} + 30 T^{3} + 24 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 40 T^{2} - 128 T^{3} + 40 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 32 T^{2} + 86 T^{3} + 32 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 14 T + 112 T^{2} + 578 T^{3} + 112 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 68 T^{2} + 182 T^{3} + 68 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 50 T^{2} + 212 T^{3} + 50 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 68 T^{2} + 400 T^{3} + 68 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 146 T^{2} - 830 T^{3} + 146 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 21 T + 212 T^{2} + 1504 T^{3} + 212 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 113 T^{2} + 124 T^{3} + 113 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 13 T + 162 T^{2} + 1362 T^{3} + 162 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 5 T + 86 T^{2} - 406 T^{3} + 86 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 17 T + 218 T^{2} + 1878 T^{3} + 218 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 13 T + 158 T^{2} + 1416 T^{3} + 158 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - T + 206 T^{2} - 134 T^{3} + 206 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 134 T^{2} - 326 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 312 T^{2} + 2632 T^{3} + 312 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 6 T + 168 T^{2} - 1018 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 11 T + 124 T^{2} + 636 T^{3} + 124 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 288 T^{2} - 1526 T^{3} + 288 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.093715837337372526674876227338, −7.61240301333667005159937732733, −7.46843508231272182759531332553, −7.31697270335692530933227410575, −6.76284324489530913474083377115, −6.76198466517805387755456818514, −6.59708236807918416555809074184, −6.07563832067743967213070055747, −6.00285926038154593434048896675, −5.97513467970114208074076757894, −5.13431762919321298078630388736, −4.92611070714413072711904277125, −4.84288898971428583755625781255, −4.28998917873607582524536316708, −4.13211505178256567157006554194, −3.98616931194630644003386509535, −3.56774557547826459412836660006, −3.51773287067907128418798778876, −3.29336515869630848053944034861, −2.74261835464045197543178945395, −2.64844926765797918249999603555, −2.26178818640748139080905645578, −1.89396719079870897524247629410, −1.44396882014989010030824755159, −1.36571280204201067351141203407, 0, 0, 0,
1.36571280204201067351141203407, 1.44396882014989010030824755159, 1.89396719079870897524247629410, 2.26178818640748139080905645578, 2.64844926765797918249999603555, 2.74261835464045197543178945395, 3.29336515869630848053944034861, 3.51773287067907128418798778876, 3.56774557547826459412836660006, 3.98616931194630644003386509535, 4.13211505178256567157006554194, 4.28998917873607582524536316708, 4.84288898971428583755625781255, 4.92611070714413072711904277125, 5.13431762919321298078630388736, 5.97513467970114208074076757894, 6.00285926038154593434048896675, 6.07563832067743967213070055747, 6.59708236807918416555809074184, 6.76198466517805387755456818514, 6.76284324489530913474083377115, 7.31697270335692530933227410575, 7.46843508231272182759531332553, 7.61240301333667005159937732733, 8.093715837337372526674876227338
