| L(s) = 1 | − 7·4-s − 32·7-s − 488·13-s − 207·16-s + 610·19-s + 1.21e3·25-s + 224·28-s + 2.96e3·31-s − 2.07e3·37-s − 716·43-s − 4.03e3·49-s + 3.41e3·52-s + 3.08e3·61-s + 3.24e3·64-s − 6.67e3·67-s − 2.03e3·73-s − 4.27e3·76-s + 3.35e3·79-s + 1.56e4·91-s + 9.56e3·97-s − 8.47e3·100-s + 1.21e4·103-s − 3.20e4·109-s + 6.62e3·112-s + 2.53e4·121-s − 2.07e4·124-s + 127-s + ⋯ |
| L(s) = 1 | − 0.437·4-s − 0.653·7-s − 2.88·13-s − 0.808·16-s + 1.68·19-s + 1.93·25-s + 2/7·28-s + 3.08·31-s − 1.51·37-s − 0.387·43-s − 1.68·49-s + 1.26·52-s + 0.829·61-s + 0.791·64-s − 1.48·67-s − 0.380·73-s − 0.739·76-s + 0.537·79-s + 1.88·91-s + 1.01·97-s − 0.847·100-s + 1.14·103-s − 2.69·109-s + 0.528·112-s + 1.73·121-s − 1.35·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8067235859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8067235859\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - 5 T + p^{4} T^{2} )( 1 + 5 T + p^{4} T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 1211 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 16 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 25382 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 244 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 108142 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 305 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 236723 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1269443 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 1484 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 28 p T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5651366 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 358 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3405517 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15763763 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 263138 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1544 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3337 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 19022411 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 1015 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1676 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 71793542 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 91361914 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4781 T + p^{4} T^{2} )^{2} \) |
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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
−11.67903433240824869689178072342, −11.49881559488635880154203896356, −10.47118114858026543218336116925, −10.04772618174316519179685188164, −9.918000388647355479616286593788, −9.312790175819873384479468185376, −8.932480050747464986524254495818, −8.274853864742640843390765245673, −7.65802121443355967749754525735, −7.17235669014793531806163941645, −6.75203372668965665347186023247, −6.28642265733393909930121508948, −5.18231735954352261978736400727, −4.82242729513335426006328135785, −4.75066464059397066258947104090, −3.56168548664986880598904893692, −2.70973431662401336328883952180, −2.63873344233151642943061381857, −1.25873469435520465119445467418, −0.31816827871012446814724682029,
0.31816827871012446814724682029, 1.25873469435520465119445467418, 2.63873344233151642943061381857, 2.70973431662401336328883952180, 3.56168548664986880598904893692, 4.75066464059397066258947104090, 4.82242729513335426006328135785, 5.18231735954352261978736400727, 6.28642265733393909930121508948, 6.75203372668965665347186023247, 7.17235669014793531806163941645, 7.65802121443355967749754525735, 8.274853864742640843390765245673, 8.932480050747464986524254495818, 9.312790175819873384479468185376, 9.918000388647355479616286593788, 10.04772618174316519179685188164, 10.47118114858026543218336116925, 11.49881559488635880154203896356, 11.67903433240824869689178072342
