| L(s) = 1 | + (−3.33 + 2.79i)2-s + (−15.5 + 0.263i)3-s + (−2.27 + 12.8i)4-s + (28.5 − 10.4i)5-s + (51.1 − 44.4i)6-s + (−33.9 − 192. i)7-s + (−98.0 − 169. i)8-s + (242. − 8.21i)9-s + (−66.1 + 114. i)10-s + (394. + 143. i)11-s + (32.0 − 201. i)12-s + (−753. − 632. i)13-s + (650. + 545. i)14-s + (−442. + 169. i)15-s + (407. + 148. i)16-s + (−370. + 641. i)17-s + ⋯ |
| L(s) = 1 | + (−0.588 + 0.494i)2-s + (−0.999 + 0.0169i)3-s + (−0.0710 + 0.403i)4-s + (0.511 − 0.186i)5-s + (0.580 − 0.503i)6-s + (−0.261 − 1.48i)7-s + (−0.541 − 0.938i)8-s + (0.999 − 0.0337i)9-s + (−0.209 + 0.362i)10-s + (0.983 + 0.358i)11-s + (0.0642 − 0.404i)12-s + (−1.23 − 1.03i)13-s + (0.887 + 0.744i)14-s + (−0.508 + 0.194i)15-s + (0.397 + 0.144i)16-s + (−0.310 + 0.538i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.415688 - 0.311888i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.415688 - 0.311888i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (15.5 - 0.263i)T \) |
| good | 2 | \( 1 + (3.33 - 2.79i)T + (5.55 - 31.5i)T^{2} \) |
| 5 | \( 1 + (-28.5 + 10.4i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (33.9 + 192. i)T + (-1.57e4 + 5.74e3i)T^{2} \) |
| 11 | \( 1 + (-394. - 143. i)T + (1.23e5 + 1.03e5i)T^{2} \) |
| 13 | \( 1 + (753. + 632. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (370. - 641. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (625. + 1.08e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-637. + 3.61e3i)T + (-6.04e6 - 2.20e6i)T^{2} \) |
| 29 | \( 1 + (318. - 267. i)T + (3.56e6 - 2.01e7i)T^{2} \) |
| 31 | \( 1 + (-1.15e3 + 6.52e3i)T + (-2.69e7 - 9.79e6i)T^{2} \) |
| 37 | \( 1 + (636. - 1.10e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (7.66e3 + 6.43e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-4.88e3 - 1.77e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-3.08e3 - 1.75e4i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 - 5.15e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (2.52e4 - 9.19e3i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-3.71e3 - 2.10e4i)T + (-7.93e8 + 2.88e8i)T^{2} \) |
| 67 | \( 1 + (7.89e3 + 6.62e3i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-2.11e4 + 3.66e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (432. + 749. i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-8.00e4 + 6.71e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (2.93e4 - 2.46e4i)T + (6.84e8 - 3.87e9i)T^{2} \) |
| 89 | \( 1 + (-4.19e4 - 7.26e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (3.18e4 + 1.16e4i)T + (6.57e9 + 5.51e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.75805353555630558214029735664, −15.15600653207501400620351665585, −13.28379972152860026961848665755, −12.34004215254713981605055219994, −10.57251349606200069059029406755, −9.523729435812660086461444259372, −7.51482804751699091590613042055, −6.48895593303786878731684223593, −4.32029065093275035181320314027, −0.46479930430982686345764736696,
1.91060477790726760434056098230, 5.29976455227028135596104426610, 6.47117868689815240045858109442, 9.110578039420277715651469715257, 9.910230842068499305182506021396, 11.51924514004532223610333977242, 12.13809057658679688828306799228, 14.14291670530872842150832202088, 15.42749956537856638956870876806, 16.90128536833506698163548635211
