| L(s) = 1 | − 6·2-s − 18·3-s − 6·4-s + 108·6-s − 102·7-s + 96·8-s + 243·9-s − 12·11-s + 108·12-s − 1.05e3·13-s + 612·14-s − 196·16-s + 1.71e3·17-s − 1.45e3·18-s + 4.21e3·19-s + 1.83e3·21-s + 72·22-s + 444·23-s − 1.72e3·24-s + 6.32e3·26-s − 2.91e3·27-s + 612·28-s + 4.06e3·29-s − 2.59e3·31-s + 4.44e3·32-s + 216·33-s − 1.02e4·34-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1.15·3-s − 0.187·4-s + 1.22·6-s − 0.786·7-s + 0.530·8-s + 9-s − 0.0299·11-s + 0.216·12-s − 1.72·13-s + 0.834·14-s − 0.191·16-s + 1.44·17-s − 1.06·18-s + 2.67·19-s + 0.908·21-s + 0.0317·22-s + 0.175·23-s − 0.612·24-s + 1.83·26-s − 0.769·27-s + 0.147·28-s + 0.898·29-s − 0.485·31-s + 0.766·32-s + 0.0345·33-s − 1.52·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5104406250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5104406250\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 3 p T + 21 p T^{2} + 3 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 102 T + 18359 T^{2} + 102 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 12 T + 82074 T^{2} + 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1054 T + 734619 T^{2} + 1054 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1716 T + 3526278 T^{2} - 1716 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4214 T + 8516703 T^{2} - 4214 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 444 T - 95054 T^{2} - 444 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4068 T + 45062238 T^{2} - 4068 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2598 T + 31786727 T^{2} + 2598 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4412 T + 45774894 T^{2} + 4412 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11232 T + 260535762 T^{2} - 11232 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8450 T + 310421175 T^{2} - 8450 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2460 T + 116859810 T^{2} + 2460 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 65064 T + 1892815386 T^{2} - 65064 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 63924 T + 2170928058 T^{2} + 63924 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7310 T + 1671053643 T^{2} - 7310 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 61734 T + 2827628303 T^{2} + 61734 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 98304 T + 4832737806 T^{2} - 98304 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 26564 T + 3962506326 T^{2} - 26564 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 84000 T + 7803834398 T^{2} - 84000 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 65772 T + 8941285738 T^{2} - 65772 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 103104 T + 13514604658 T^{2} - 103104 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 69374 T + 3264550083 T^{2} + 69374 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−13.87116178366219855943605077291, −13.22556910490553649846558720122, −12.31176551697321227518576350021, −12.18044059166172331914661079117, −11.81911021568275022796946700605, −10.90558253342752815129862765728, −10.16576743182755240288151917012, −9.937095167372252681944596318178, −9.381445348221014754590280785713, −9.105384672710450502470991472878, −7.69671035119999400620446984594, −7.69211878025834467820361292235, −6.90071364034252217168997420029, −6.03260866154167651105957285519, −5.23023619807702693043973413150, −4.94419692492687297011832114783, −3.66799091153197237441082371214, −2.72466489868183062983644799971, −1.04977524744511636072588525309, −0.51785399122690011171773288818,
0.51785399122690011171773288818, 1.04977524744511636072588525309, 2.72466489868183062983644799971, 3.66799091153197237441082371214, 4.94419692492687297011832114783, 5.23023619807702693043973413150, 6.03260866154167651105957285519, 6.90071364034252217168997420029, 7.69211878025834467820361292235, 7.69671035119999400620446984594, 9.105384672710450502470991472878, 9.381445348221014754590280785713, 9.937095167372252681944596318178, 10.16576743182755240288151917012, 10.90558253342752815129862765728, 11.81911021568275022796946700605, 12.18044059166172331914661079117, 12.31176551697321227518576350021, 13.22556910490553649846558720122, 13.87116178366219855943605077291
