| L(s) = 1 | + 38·5-s − 294·7-s − 130·11-s − 112·13-s + 2.58e3·17-s + 1.92e3·19-s − 4.87e3·23-s + 1.49e3·25-s − 1.54e3·29-s + 2.80e3·31-s − 1.11e4·35-s + 4.42e3·37-s − 5.65e3·41-s + 9.47e3·43-s − 1.40e4·47-s + 3.78e4·49-s + 1.57e4·53-s − 4.94e3·55-s + 3.37e4·59-s + 9.62e4·61-s − 4.25e3·65-s + 8.00e4·67-s − 1.20e5·71-s + 1.47e4·73-s + 3.82e4·77-s + 2.77e4·79-s − 1.41e5·83-s + ⋯ |
| L(s) = 1 | + 0.679·5-s − 2.26·7-s − 0.323·11-s − 0.183·13-s + 2.16·17-s + 1.22·19-s − 1.92·23-s + 0.477·25-s − 0.341·29-s + 0.524·31-s − 1.54·35-s + 0.531·37-s − 0.525·41-s + 0.781·43-s − 0.926·47-s + 2.25·49-s + 0.771·53-s − 0.220·55-s + 1.26·59-s + 3.31·61-s − 0.124·65-s + 2.17·67-s − 2.83·71-s + 0.324·73-s + 0.734·77-s + 0.500·79-s − 2.25·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.923883235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923883235\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 38 T - 49 T^{2} - 38 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 p^{2} T + 48563 T^{2} + 6 p^{7} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 130 T + 86567 T^{2} + 130 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 112 T + 55314 p T^{2} + 112 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 152 p T + 4482338 T^{2} - 152 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1924 T + 4172682 T^{2} - 1924 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 212 p T + 17511170 T^{2} + 212 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1548 T + 30965374 T^{2} + 1548 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2806 T - 17023629 T^{2} - 2806 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4428 T + 63003710 T^{2} - 4428 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5652 T + 232878838 T^{2} + 5652 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9476 T + 105450090 T^{2} - 9476 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14028 T + 424343170 T^{2} + 14028 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15770 T - 48014929 T^{2} - 15770 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 33752 T + 1368434534 T^{2} - 33752 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 96224 T + 3914340186 T^{2} - 96224 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 80020 T + 3884800314 T^{2} - 80020 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 120296 T + 7024602446 T^{2} + 120296 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14762 T + 2922941307 T^{2} - 14762 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 27752 T + 6343432734 T^{2} - 27752 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 141362 T + 11501152487 T^{2} + 141362 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 211452 T + 22334357734 T^{2} - 211452 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 53762 T + 10243703235 T^{2} + 53762 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.71747982176847281584906308101, −11.39943330861995300098470681271, −10.16741606836434318416111519856, −10.12273404192632525655558101084, −9.861379517063237705208935335065, −9.604819770832283153787938969349, −8.797296758730171371702787240045, −8.229940091069116329931580684211, −7.52930594514566468457553132269, −7.18835316194470960914586074291, −6.38497279557453705035799596929, −6.10143712606375861489644878854, −5.51200657571230554638243226165, −5.13452286529969255096395164786, −3.78518091736232601728397613443, −3.62045875495340351840231595224, −2.85559894370451402513646957239, −2.30826816474676552961320818801, −1.14371097143906285294088259114, −0.46106019138221144436394998398,
0.46106019138221144436394998398, 1.14371097143906285294088259114, 2.30826816474676552961320818801, 2.85559894370451402513646957239, 3.62045875495340351840231595224, 3.78518091736232601728397613443, 5.13452286529969255096395164786, 5.51200657571230554638243226165, 6.10143712606375861489644878854, 6.38497279557453705035799596929, 7.18835316194470960914586074291, 7.52930594514566468457553132269, 8.229940091069116329931580684211, 8.797296758730171371702787240045, 9.604819770832283153787938969349, 9.861379517063237705208935335065, 10.12273404192632525655558101084, 10.16741606836434318416111519856, 11.39943330861995300098470681271, 11.71747982176847281584906308101
