| L(s) = 1 | − 6·2-s + 24·4-s − 20·5-s + 24·7-s − 80·8-s + 120·10-s − 10·11-s − 4·13-s − 144·14-s + 240·16-s + 66·17-s − 164·19-s − 480·20-s + 60·22-s + 204·23-s + 52·25-s + 24·26-s + 576·28-s − 87·29-s − 86·31-s − 672·32-s − 396·34-s − 480·35-s − 42·37-s + 984·38-s + 1.60e3·40-s − 562·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 1.78·5-s + 1.29·7-s − 3.53·8-s + 3.79·10-s − 0.274·11-s − 0.0853·13-s − 2.74·14-s + 15/4·16-s + 0.941·17-s − 1.98·19-s − 5.36·20-s + 0.581·22-s + 1.84·23-s + 0.415·25-s + 0.181·26-s + 3.88·28-s − 0.557·29-s − 0.498·31-s − 3.71·32-s − 1.99·34-s − 2.31·35-s − 0.186·37-s + 4.20·38-s + 6.32·40-s − 2.14·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 4 p T + 348 T^{2} + 4838 T^{3} + 348 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 24 T + 533 T^{2} - 1584 p T^{3} + 533 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 3760 T^{2} + 24196 T^{3} + 3760 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 972 T^{2} + 149282 T^{3} + 972 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 66 T + 4079 T^{2} + 30852 T^{3} + 4079 p^{3} T^{4} - 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 164 T + 20453 T^{2} + 1585304 T^{3} + 20453 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 204 T + 40785 T^{2} - 4286760 T^{3} + 40785 p^{3} T^{4} - 204 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 86 T + 41284 T^{2} + 357880 T^{3} + 41284 p^{3} T^{4} + 86 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 42 T + 72503 T^{2} - 3430044 T^{3} + 72503 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 562 T + 309195 T^{2} + 83449252 T^{3} + 309195 p^{3} T^{4} + 562 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 18 T + 228152 T^{2} - 2665764 T^{3} + 228152 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 654 T + 409484 T^{2} + 139225608 T^{3} + 409484 p^{3} T^{4} + 654 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 712 T + 96060 T^{2} - 40119698 T^{3} + 96060 p^{3} T^{4} + 712 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 184 T + 304693 T^{2} + 18216544 T^{3} + 304693 p^{3} T^{4} + 184 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 322 T + 292571 T^{2} - 151430188 T^{3} + 292571 p^{3} T^{4} - 322 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 228 T + 574449 T^{2} + 184756120 T^{3} + 574449 p^{3} T^{4} + 228 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 52 T + 1073569 T^{2} - 37222072 T^{3} + 1073569 p^{3} T^{4} - 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 494 T + 1173163 T^{2} + 374938588 T^{3} + 1173163 p^{3} T^{4} + 494 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2110 T + 2879660 T^{2} + 2365811752 T^{3} + 2879660 p^{3} T^{4} + 2110 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 288 T + 676365 T^{2} + 108256752 T^{3} + 676365 p^{3} T^{4} - 288 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 914 T + 1450107 T^{2} + 690671780 T^{3} + 1450107 p^{3} T^{4} + 914 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 218 T + 1237251 T^{2} - 414931540 T^{3} + 1237251 p^{3} T^{4} - 218 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.713988494049806162336174324219, −9.407390655746228931128102630610, −9.007204890178900121320425195839, −8.691062656434793407875731877800, −8.332397409839980588469119199385, −8.265082369996408960361029758449, −8.202265356605173528223512409410, −7.54047831727388235921365129336, −7.49470896502454386092065610437, −7.43089661223318337287165644663, −6.68813633188779944391538307530, −6.56980887591608105413741580182, −6.40482560135211149216664033269, −5.62347541953503055320372047919, −5.25687322031443145039940767721, −5.10173122618873332751516727052, −4.56087478416973895712170735827, −4.00583656917212256088200945764, −3.93405933755530498601890259173, −3.16779593391606924677036981987, −3.02865890544754311347787824013, −2.48536414219912310704082059344, −1.75597322770101554756201940031, −1.42923750365593147893384171056, −1.29096782322411110448415981178, 0, 0, 0,
1.29096782322411110448415981178, 1.42923750365593147893384171056, 1.75597322770101554756201940031, 2.48536414219912310704082059344, 3.02865890544754311347787824013, 3.16779593391606924677036981987, 3.93405933755530498601890259173, 4.00583656917212256088200945764, 4.56087478416973895712170735827, 5.10173122618873332751516727052, 5.25687322031443145039940767721, 5.62347541953503055320372047919, 6.40482560135211149216664033269, 6.56980887591608105413741580182, 6.68813633188779944391538307530, 7.43089661223318337287165644663, 7.49470896502454386092065610437, 7.54047831727388235921365129336, 8.202265356605173528223512409410, 8.265082369996408960361029758449, 8.332397409839980588469119199385, 8.691062656434793407875731877800, 9.007204890178900121320425195839, 9.407390655746228931128102630610, 9.713988494049806162336174324219
