| L(s) = 1 | + 6·2-s − 10·3-s + 24·4-s − 12·5-s − 60·6-s − 13·7-s + 80·8-s + 29·9-s − 72·10-s + 33·11-s − 240·12-s − 55·13-s − 78·14-s + 120·15-s + 240·16-s − 32·17-s + 174·18-s + 57·19-s − 288·20-s + 130·21-s + 198·22-s − 381·23-s − 800·24-s − 123·25-s − 330·26-s + 61·27-s − 312·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.92·3-s + 3·4-s − 1.07·5-s − 4.08·6-s − 0.701·7-s + 3.53·8-s + 1.07·9-s − 2.27·10-s + 0.904·11-s − 5.77·12-s − 1.17·13-s − 1.48·14-s + 2.06·15-s + 15/4·16-s − 0.456·17-s + 2.27·18-s + 0.688·19-s − 3.21·20-s + 1.35·21-s + 1.91·22-s − 3.45·23-s − 6.80·24-s − 0.983·25-s − 2.48·26-s + 0.434·27-s − 2.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73034632 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73034632 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 10 T + 71 T^{2} + 359 T^{3} + 71 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 12 T + 267 T^{2} + 1731 T^{3} + 267 p^{3} T^{4} + 12 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 13 T + 800 T^{2} + 5932 T^{3} + 800 p^{3} T^{4} + 13 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 55 T + 6516 T^{2} + 237170 T^{3} + 6516 p^{3} T^{4} + 55 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 32 T + 547 p T^{2} + 285576 T^{3} + 547 p^{4} T^{4} + 32 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 381 T + 78063 T^{2} + 10420658 T^{3} + 78063 p^{3} T^{4} + 381 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 207 T + 2400 p T^{2} + 10111826 T^{3} + 2400 p^{4} T^{4} + 207 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 86177 T^{2} - 252553 T^{3} + 86177 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 181 T + 82959 T^{2} + 10805186 T^{3} + 82959 p^{3} T^{4} + 181 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 671 T + 251942 T^{2} + 70610754 T^{3} + 251942 p^{3} T^{4} + 671 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 399 T + 288552 T^{2} + 65458604 T^{3} + 288552 p^{3} T^{4} + 399 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 80 T + 245225 T^{2} + 21127488 T^{3} + 245225 p^{3} T^{4} + 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 48 T + 167919 T^{2} - 3730200 T^{3} + 167919 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 119 T + 438153 T^{2} - 49269722 T^{3} + 438153 p^{3} T^{4} - 119 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 500 T + 645043 T^{2} - 227540392 T^{3} + 645043 p^{3} T^{4} - 500 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 1346 T + 1410937 T^{2} + 846086239 T^{3} + 1410937 p^{3} T^{4} + 1346 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 148 T + 770659 T^{2} - 29398729 T^{3} + 770659 p^{3} T^{4} - 148 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 338 T + 919783 T^{2} + 290882596 T^{3} + 919783 p^{3} T^{4} + 338 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 p T + 1096669 T^{2} + 371418752 T^{3} + 1096669 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1121 T + 2059120 T^{2} - 1299824144 T^{3} + 2059120 p^{3} T^{4} - 1121 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 481 T + 622729 T^{2} - 1173163582 T^{3} + 622729 p^{3} T^{4} - 481 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2499 T + 4524699 T^{2} + 4911234550 T^{3} + 4524699 p^{3} T^{4} + 2499 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−10.27629569550487868032045560633, −9.804206878872725286215639262428, −9.704527225823969536934572493550, −9.448343116610206086340503063364, −8.598698951797886051584031407441, −8.319496510701549167670863841225, −8.046627928262083619524879433706, −7.56430911010437659074911664733, −7.41085470437637822665562540224, −6.93177362430009808016025878881, −6.59276781672891904612460403741, −6.35424774582781350565760829235, −6.23526180844467205760186564925, −5.59593300862518948613561775038, −5.47291415550930373215221642082, −5.39731770601757821108066230900, −4.67197655979051600789253692687, −4.51305822521773155137198332713, −4.10941471512178066871146135356, −3.72678906579797027011242921894, −3.32443771178295250282754494248, −3.24816122338716250451876574503, −2.26518603863165701979591683209, −1.88068279557373200100614254361, −1.50056010418548717121262751733, 0, 0, 0,
1.50056010418548717121262751733, 1.88068279557373200100614254361, 2.26518603863165701979591683209, 3.24816122338716250451876574503, 3.32443771178295250282754494248, 3.72678906579797027011242921894, 4.10941471512178066871146135356, 4.51305822521773155137198332713, 4.67197655979051600789253692687, 5.39731770601757821108066230900, 5.47291415550930373215221642082, 5.59593300862518948613561775038, 6.23526180844467205760186564925, 6.35424774582781350565760829235, 6.59276781672891904612460403741, 6.93177362430009808016025878881, 7.41085470437637822665562540224, 7.56430911010437659074911664733, 8.046627928262083619524879433706, 8.319496510701549167670863841225, 8.598698951797886051584031407441, 9.448343116610206086340503063364, 9.704527225823969536934572493550, 9.804206878872725286215639262428, 10.27629569550487868032045560633
