| L(s) = 1 | − 3·2-s − 4-s − 15·5-s + 7·7-s + 9·8-s + 45·10-s − 66·11-s − 11·13-s − 21·14-s − 9·16-s − 99·17-s − 77·19-s + 15·20-s + 198·22-s − 33·23-s − 73·25-s + 33·26-s − 7·28-s + 51·29-s + 43·31-s + 153·32-s + 297·34-s − 105·35-s − 50·37-s + 231·38-s − 135·40-s − 132·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1/8·4-s − 1.34·5-s + 0.377·7-s + 0.397·8-s + 1.42·10-s − 1.80·11-s − 0.234·13-s − 0.400·14-s − 0.140·16-s − 1.41·17-s − 0.929·19-s + 0.167·20-s + 1.91·22-s − 0.299·23-s − 0.583·25-s + 0.248·26-s − 0.0472·28-s + 0.326·29-s + 0.249·31-s + 0.845·32-s + 1.49·34-s − 0.507·35-s − 0.222·37-s + 0.986·38-s − 0.533·40-s − 0.502·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 + 3 T + 5 p T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 p T + 298 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - p T + 624 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 p T + 293 p T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 11 T + 2568 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 99 T + 11608 T^{2} + 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 77 T + 9186 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 33 T + 21628 T^{2} + 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 51 T + 49420 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 43 T + 59970 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 50 T + 77874 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 132 T + 62965 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 88 T + 160653 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 399 T + 179128 T^{2} + 399 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 54 T + 81970 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 798 T + 417367 T^{2} + 798 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 439 T + 411186 T^{2} - 439 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 988 T + 809625 T^{2} - 988 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1368 T + 1012606 T^{2} + 1368 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 455 T + 342636 T^{2} + 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 803 T + 359562 T^{2} + 803 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 813 T + 1214362 T^{2} + 813 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 396 T + 1406374 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 736 T + 1874937 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.27821050672043648651569477118, −13.22108481913971931631577836250, −12.48645682542639537645263523440, −11.77377589333832168646177016946, −11.27989395502828336414464370008, −10.89513760914469573537954295692, −10.01524481259845226850486140110, −9.889093309361191149801700439886, −8.674933380711729500907866632007, −8.621861034883653846468964614760, −7.87727581625584982046037480331, −7.67062842802970788318753774702, −6.73927698132108676157735784772, −5.86024017939662218629439332761, −4.63075146327798786792425699523, −4.53682533596965237153114113345, −3.25309316451567943879791015046, −2.10838781592008252593949288477, 0, 0,
2.10838781592008252593949288477, 3.25309316451567943879791015046, 4.53682533596965237153114113345, 4.63075146327798786792425699523, 5.86024017939662218629439332761, 6.73927698132108676157735784772, 7.67062842802970788318753774702, 7.87727581625584982046037480331, 8.621861034883653846468964614760, 8.674933380711729500907866632007, 9.889093309361191149801700439886, 10.01524481259845226850486140110, 10.89513760914469573537954295692, 11.27989395502828336414464370008, 11.77377589333832168646177016946, 12.48645682542639537645263523440, 13.22108481913971931631577836250, 13.27821050672043648651569477118
