| L(s) = 1 | − 3·2-s − 3·3-s + 10·4-s − 15·5-s + 9·6-s − 7·7-s − 15·8-s + 27·9-s + 45·10-s − 66·11-s − 30·12-s + 11·13-s + 21·14-s + 45·15-s + 37·16-s + 198·17-s − 81·18-s − 154·19-s − 150·20-s + 21·21-s + 198·22-s − 33·23-s + 45·24-s + 298·25-s − 33·26-s − 216·27-s − 70·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.577·3-s + 5/4·4-s − 1.34·5-s + 0.612·6-s − 0.377·7-s − 0.662·8-s + 9-s + 1.42·10-s − 1.80·11-s − 0.721·12-s + 0.234·13-s + 0.400·14-s + 0.774·15-s + 0.578·16-s + 2.82·17-s − 1.06·18-s − 1.85·19-s − 1.67·20-s + 0.218·21-s + 1.91·22-s − 0.299·23-s + 0.382·24-s + 2.38·25-s − 0.248·26-s − 1.53·27-s − 0.472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2710451220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2710451220\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T - 2 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 3 T - T^{2} - 9 p T^{3} - 9 p^{2} T^{4} - 9 p^{4} T^{5} - p^{6} T^{6} + 3 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 3 p T - 73 T^{2} + 144 p T^{3} + 45054 T^{4} + 144 p^{4} T^{5} - 73 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + p T - 575 T^{2} - 62 p T^{3} + 254920 T^{4} - 62 p^{4} T^{5} - 575 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 p T + 103 p T^{2} + 306 p^{2} T^{3} + 23292 p^{2} T^{4} + 306 p^{5} T^{5} + 103 p^{7} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 11 T - 2447 T^{2} + 20086 T^{3} + 1501978 T^{4} + 20086 p^{3} T^{5} - 2447 p^{6} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 99 T + 11608 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 77 T + 9186 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 33 T - 893 p T^{2} - 89298 T^{3} + 306484632 T^{4} - 89298 p^{3} T^{5} - 893 p^{7} T^{6} + 33 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 51 T - 46819 T^{2} - 32742 T^{3} + 1784077290 T^{4} - 32742 p^{3} T^{5} - 46819 p^{6} T^{6} - 51 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 43 T - 58121 T^{2} + 16684 T^{3} + 2653813660 T^{4} + 16684 p^{3} T^{5} - 58121 p^{6} T^{6} + 43 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 50 T + 77874 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 132 T - 45541 T^{2} - 9883764 T^{3} - 1986392520 T^{4} - 9883764 p^{3} T^{5} - 45541 p^{6} T^{6} + 132 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 88 T - 152909 T^{2} + 144232 T^{3} + 18872321152 T^{4} + 144232 p^{3} T^{5} - 152909 p^{6} T^{6} + 88 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 399 T - 19927 T^{2} - 11378682 T^{3} + 4778899632 T^{4} - 11378682 p^{3} T^{5} - 19927 p^{6} T^{6} + 399 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 54 T + 81970 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 798 T + 219437 T^{2} + 5273982 T^{3} + 1228510332 T^{4} + 5273982 p^{3} T^{5} + 219437 p^{6} T^{6} + 798 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 439 T - 218465 T^{2} - 18778664 T^{3} + 73809546934 T^{4} - 18778664 p^{3} T^{5} - 218465 p^{6} T^{6} + 439 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 988 T + 166519 T^{2} + 205601812 T^{3} + 271446260584 T^{4} + 205601812 p^{3} T^{5} + 166519 p^{6} T^{6} + 988 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1368 T + 1012606 T^{2} - 1368 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 455 T + 342636 T^{2} + 455 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 803 T + 285247 T^{2} + 503092348 T^{3} - 431718608228 T^{4} + 503092348 p^{3} T^{5} + 285247 p^{6} T^{6} - 803 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 813 T - 553393 T^{2} + 57550644 T^{3} + 769801212072 T^{4} + 57550644 p^{3} T^{5} - 553393 p^{6} T^{6} + 813 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 396 T + 1406374 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 736 T - 1333241 T^{2} + 36498976 T^{3} + 2188025435632 T^{4} + 36498976 p^{3} T^{5} - 1333241 p^{6} T^{6} + 736 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.53500639267541014152739849537, −15.58090525747337205004237575777, −15.57137665890157673680315591470, −15.45793030552983775794686496833, −14.95846146462468485433937781861, −14.31845763182110684557309037513, −13.86509770949156527156030336576, −13.03963571238007264459722957608, −12.79102408508266130325545875572, −12.45166949290232207508515373311, −12.02983002008330811332268236047, −11.63045977622110454064495631380, −10.95181254561589137292804741136, −10.55020127415590808373153523460, −10.35636168919633159210433083573, −10.02099796895845103617255852559, −9.256610890634567397347525655757, −8.338707171049150906418840569156, −8.085664835665595463502938803439, −7.57820955445575704468383890105, −7.23425613183800289541388769024, −6.42275905042016689314401269882, −5.61571502902450384024731321480, −4.67133977662776025619376926210, −3.34962024508304345866916773268,
3.34962024508304345866916773268, 4.67133977662776025619376926210, 5.61571502902450384024731321480, 6.42275905042016689314401269882, 7.23425613183800289541388769024, 7.57820955445575704468383890105, 8.085664835665595463502938803439, 8.338707171049150906418840569156, 9.256610890634567397347525655757, 10.02099796895845103617255852559, 10.35636168919633159210433083573, 10.55020127415590808373153523460, 10.95181254561589137292804741136, 11.63045977622110454064495631380, 12.02983002008330811332268236047, 12.45166949290232207508515373311, 12.79102408508266130325545875572, 13.03963571238007264459722957608, 13.86509770949156527156030336576, 14.31845763182110684557309037513, 14.95846146462468485433937781861, 15.45793030552983775794686496833, 15.57137665890157673680315591470, 15.58090525747337205004237575777, 16.53500639267541014152739849537
