| L(s) = 1 | + 3·2-s + 8·4-s − 3·5-s + 10·7-s + 45·8-s − 9·10-s + 12·11-s − 8·13-s + 30·14-s + 135·16-s + 252·17-s − 182·19-s − 24·20-s + 36·22-s − 93·23-s + 125·25-s − 24·26-s + 80·28-s + 183·29-s − 170·31-s + 360·32-s + 756·34-s − 30·35-s − 344·37-s − 546·38-s − 135·40-s + 474·41-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 4-s − 0.268·5-s + 0.539·7-s + 1.98·8-s − 0.284·10-s + 0.328·11-s − 0.170·13-s + 0.572·14-s + 2.10·16-s + 3.59·17-s − 2.19·19-s − 0.268·20-s + 0.348·22-s − 0.843·23-s + 25-s − 0.181·26-s + 0.539·28-s + 1.17·29-s − 0.984·31-s + 1.98·32-s + 3.81·34-s − 0.144·35-s − 1.52·37-s − 2.33·38-s − 0.533·40-s + 1.80·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.918839126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.918839126\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T - 116 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T - 243 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T - 1187 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T - 2133 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 93 T - 3518 T^{2} + 93 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 183 T + 9100 T^{2} - 183 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 170 T - 891 T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 172 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 474 T + 155755 T^{2} - 474 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 p T - 27 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 159 T - 78542 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 603 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 564 T + 112717 T^{2} - 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 430 T - 42081 T^{2} - 430 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 439 T - 108042 T^{2} - 439 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 351 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 727 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1232 T + 1024785 T^{2} + 1232 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 p T - 47 p^{2} T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1044 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1151 T + 412128 T^{2} + 1151 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.12300019189486292944673213952, −11.51686183192893826942957186412, −10.93934598023246544615518389976, −10.60259105446062385164616713454, −10.08925743296854688291596825647, −9.878258938350828252050349247105, −8.791999215113232876106818481499, −8.321344706607845855968826925211, −7.67602424602062479947328061508, −7.60097657099784737559318918659, −6.88406103778927268893564793427, −6.14580214909821217866223809117, −5.74550482102897005108678623734, −4.96451408103840569647013882537, −4.70110552167717228795227710964, −3.82794583630053966988711454497, −3.54203252753275120932899653215, −2.54045108392484304725486646812, −1.66124286184347842859808094257, −0.993291985078408122561129238721,
0.993291985078408122561129238721, 1.66124286184347842859808094257, 2.54045108392484304725486646812, 3.54203252753275120932899653215, 3.82794583630053966988711454497, 4.70110552167717228795227710964, 4.96451408103840569647013882537, 5.74550482102897005108678623734, 6.14580214909821217866223809117, 6.88406103778927268893564793427, 7.60097657099784737559318918659, 7.67602424602062479947328061508, 8.321344706607845855968826925211, 8.791999215113232876106818481499, 9.878258938350828252050349247105, 10.08925743296854688291596825647, 10.60259105446062385164616713454, 10.93934598023246544615518389976, 11.51686183192893826942957186412, 12.12300019189486292944673213952
