| L(s) = 1 | − 3·2-s − 465·4-s + 1.98e3·5-s − 3.69e3·7-s + 3.41e3·8-s − 5.94e3·10-s + 1.68e4·11-s + 1.16e5·13-s + 1.10e4·14-s − 3.43e4·16-s + 1.01e6·17-s − 1.52e4·19-s − 9.22e5·20-s − 5.05e4·22-s + 2.92e6·23-s + 1.03e5·25-s − 3.50e5·26-s + 1.71e6·28-s + 5.76e6·29-s − 6.57e6·31-s − 1.85e6·32-s − 3.04e6·34-s − 7.32e6·35-s − 1.16e7·37-s + 4.56e4·38-s + 6.77e6·40-s − 2.22e7·41-s + ⋯ |
| L(s) = 1 | − 0.132·2-s − 0.908·4-s + 1.41·5-s − 0.581·7-s + 0.294·8-s − 0.188·10-s + 0.347·11-s + 1.13·13-s + 0.0770·14-s − 0.130·16-s + 2.94·17-s − 0.0267·19-s − 1.28·20-s − 0.0460·22-s + 2.18·23-s + 0.0529·25-s − 0.150·26-s + 0.527·28-s + 1.51·29-s − 1.27·31-s − 0.312·32-s − 0.390·34-s − 0.824·35-s − 1.02·37-s + 0.00355·38-s + 0.418·40-s − 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19683 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19683 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.686480845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.686480845\) |
| \(L(\frac{11}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 + 3 T + 237 p T^{2} - 75 p^{3} T^{3} + 237 p^{10} T^{4} + 3 p^{18} T^{5} + p^{27} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 1983 T + 3828966 T^{2} - 156807831 p^{2} T^{3} + 3828966 p^{9} T^{4} - 1983 p^{18} T^{5} + p^{27} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3693 T + 8268972 p T^{2} + 7926812473 p^{2} T^{3} + 8268972 p^{10} T^{4} + 3693 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 1533 p T + 4577898156 T^{2} - 109970589171831 T^{3} + 4577898156 p^{9} T^{4} - 1533 p^{19} T^{5} + p^{27} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 116916 T + 15039711735 T^{2} - 1523717549995016 T^{3} + 15039711735 p^{9} T^{4} - 116916 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 1014048 T + 618168912243 T^{2} - 14610762415245888 p T^{3} + 618168912243 p^{9} T^{4} - 1014048 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 15222 T + 368176140789 T^{2} + 14286898979746540 T^{3} + 368176140789 p^{9} T^{4} + 15222 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 127266 p T + 4468601755185 T^{2} - 5974885574099508396 T^{3} + 4468601755185 p^{9} T^{4} - 127266 p^{19} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5768790 T + 47258789748819 T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + 47258789748819 p^{9} T^{4} - 5768790 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6575223 T + 61568816990820 T^{2} + \)\(28\!\cdots\!35\)\( T^{3} + 61568816990820 p^{9} T^{4} + 6575223 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 11686026 T + 301605126050715 T^{2} + \)\(19\!\cdots\!24\)\( T^{3} + 301605126050715 p^{9} T^{4} + 11686026 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 22213518 T + 855448699464951 T^{2} + \)\(10\!\cdots\!16\)\( T^{3} + 855448699464951 p^{9} T^{4} + 22213518 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 45384414 T + 584783318055165 T^{2} + 73546225860867061156 T^{3} + 584783318055165 p^{9} T^{4} - 45384414 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12392034 T + 2691608802652473 T^{2} - \)\(17\!\cdots\!96\)\( T^{3} + 2691608802652473 p^{9} T^{4} - 12392034 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 80579637 T + 8930124888804942 T^{2} - \)\(39\!\cdots\!45\)\( T^{3} + 8930124888804942 p^{9} T^{4} - 80579637 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 244026660 T + 36787781631518529 T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + 36787781631518529 p^{9} T^{4} - 244026660 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 369729960 T + 75523544279828439 T^{2} - \)\(98\!\cdots\!68\)\( T^{3} + 75523544279828439 p^{9} T^{4} - 369729960 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 252614586 T + 83961031654694373 T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + 83961031654694373 p^{9} T^{4} + 252614586 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 403193088 T + 169497868047009621 T^{2} - \)\(35\!\cdots\!16\)\( T^{3} + 169497868047009621 p^{9} T^{4} - 403193088 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 5570229 p T + 211748488417949982 T^{2} + \)\(47\!\cdots\!97\)\( T^{3} + 211748488417949982 p^{9} T^{4} + 5570229 p^{19} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 265451856 T + 192162710472400941 T^{2} - \)\(55\!\cdots\!20\)\( T^{3} + 192162710472400941 p^{9} T^{4} - 265451856 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 121625871 T + 521691450587877156 T^{2} - \)\(43\!\cdots\!19\)\( T^{3} + 521691450587877156 p^{9} T^{4} - 121625871 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 377904006 T + 336010266400343559 T^{2} - \)\(36\!\cdots\!08\)\( T^{3} + 336010266400343559 p^{9} T^{4} - 377904006 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 438907539 T + 2149754009370295110 T^{2} + \)\(64\!\cdots\!71\)\( T^{3} + 2149754009370295110 p^{9} T^{4} + 438907539 p^{18} T^{5} + p^{27} T^{6} \) |
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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
−13.78268760694138106283696087720, −13.07486950324238025596870580072, −12.87906360041761217987777188142, −12.60186324098280595748284169498, −11.75237727851274496538554588559, −11.55627069486570946677737815058, −10.68189444762399446408471620669, −10.36306072555061274921676127162, −9.816588271234472183106361359689, −9.724076735022781039898880468487, −9.084182242151970970500206929957, −8.576929223328281741537800733786, −8.425473719299466482171402438826, −7.26428683658589054613712740796, −7.15873236176367245838248939618, −6.25879986545126302621323228229, −5.80849503420243861638131854253, −5.28900321543201426210448343409, −4.98288922584070148991954436779, −3.66208161745196348406967059103, −3.64870625246161787717714514855, −2.69844860864517553961576996314, −1.79777596707653475781068263878, −0.900555849938266409553346313049, −0.841400463414997880981647575832,
0.841400463414997880981647575832, 0.900555849938266409553346313049, 1.79777596707653475781068263878, 2.69844860864517553961576996314, 3.64870625246161787717714514855, 3.66208161745196348406967059103, 4.98288922584070148991954436779, 5.28900321543201426210448343409, 5.80849503420243861638131854253, 6.25879986545126302621323228229, 7.15873236176367245838248939618, 7.26428683658589054613712740796, 8.425473719299466482171402438826, 8.576929223328281741537800733786, 9.084182242151970970500206929957, 9.724076735022781039898880468487, 9.816588271234472183106361359689, 10.36306072555061274921676127162, 10.68189444762399446408471620669, 11.55627069486570946677737815058, 11.75237727851274496538554588559, 12.60186324098280595748284169498, 12.87906360041761217987777188142, 13.07486950324238025596870580072, 13.78268760694138106283696087720
