L(s) = 1 | + (−0.883 + 0.741i)2-s + (−5.10 + 0.962i)3-s + (−1.15 + 6.56i)4-s + (−7.21 + 2.62i)5-s + (3.79 − 4.63i)6-s + (2.25 + 12.7i)7-s + (−8.46 − 14.6i)8-s + (25.1 − 9.82i)9-s + (4.42 − 7.67i)10-s + (−10.9 − 3.99i)11-s + (−0.407 − 34.6i)12-s + (54.8 + 46.0i)13-s + (−11.4 − 9.61i)14-s + (34.3 − 20.3i)15-s + (−31.7 − 11.5i)16-s + (−18.1 + 31.4i)17-s + ⋯ |
L(s) = 1 | + (−0.312 + 0.262i)2-s + (−0.982 + 0.185i)3-s + (−0.144 + 0.820i)4-s + (−0.645 + 0.234i)5-s + (0.258 − 0.315i)6-s + (0.121 + 0.689i)7-s + (−0.373 − 0.647i)8-s + (0.931 − 0.364i)9-s + (0.140 − 0.242i)10-s + (−0.300 − 0.109i)11-s + (−0.00979 − 0.833i)12-s + (1.16 + 0.981i)13-s + (−0.218 − 0.183i)14-s + (0.590 − 0.350i)15-s + (−0.496 − 0.180i)16-s + (−0.258 + 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.211738 + 0.515667i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.211738 + 0.515667i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (5.10 - 0.962i)T \) |
good | 2 | \( 1 + (0.883 - 0.741i)T + (1.38 - 7.87i)T^{2} \) |
| 5 | \( 1 + (7.21 - 2.62i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-2.25 - 12.7i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (10.9 + 3.99i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-54.8 - 46.0i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (18.1 - 31.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (11.8 + 20.5i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (34.5 - 196. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-218. + 183. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-5.40 + 30.6i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-171. + 296. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-188. - 158. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (143. + 52.3i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-49.8 - 282. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 321.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (382. - 139. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-36.2 - 205. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (152. + 127. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (82.6 - 143. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-484. - 838. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-759. + 637. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-440. + 369. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (140. + 243. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-446. - 162. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−17.35038290433287415225842146211, −16.02009753312903360268546954330, −15.53588902079240456704400532047, −13.34469067144779348811851830116, −11.96947591199880727492625500988, −11.21997421576208337565995290066, −9.273312771289978362395807645424, −7.77298226708569010748330096188, −6.19787277208682350758293250680, −4.03233353353448226112015909150,
0.71653981401384729909148117373, 4.71778786158556375403026532737, 6.37121636747613915891305958555, 8.248567670133603496136051549933, 10.29577896762193454498923990624, 10.96124387966262686905612682403, 12.36492921282942078955564771609, 13.79486900884739976579409173593, 15.43812185960225948607248059498, 16.42249300166516496351507503810