| L(s) = 1 | + (1.01 + 1.40i)2-s + (−1.60 − 4.94i)3-s + (4.01 − 12.3i)4-s + (−32.0 − 23.2i)5-s + (5.29 − 7.28i)6-s + (27.3 + 8.87i)7-s + (47.8 − 15.5i)8-s + (−21.8 + 15.8i)9-s − 68.5i·10-s + (30.6 − 117. i)11-s − 67.5·12-s + (94.6 + 130. i)13-s + (15.3 + 47.3i)14-s + (−63.5 + 195. i)15-s + (−97.7 − 71.0i)16-s + (−141. + 195. i)17-s + ⋯ |
| L(s) = 1 | + (0.254 + 0.350i)2-s + (−0.178 − 0.549i)3-s + (0.250 − 0.772i)4-s + (−1.28 − 0.930i)5-s + (0.147 − 0.202i)6-s + (0.557 + 0.181i)7-s + (0.746 − 0.242i)8-s + (−0.269 + 0.195i)9-s − 0.685i·10-s + (0.252 − 0.967i)11-s − 0.468·12-s + (0.559 + 0.770i)13-s + (0.0784 + 0.241i)14-s + (−0.282 + 0.868i)15-s + (−0.381 − 0.277i)16-s + (−0.490 + 0.675i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.01361 - 0.891522i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01361 - 0.891522i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.60 + 4.94i)T \) |
| 11 | \( 1 + (-30.6 + 117. i)T \) |
| good | 2 | \( 1 + (-1.01 - 1.40i)T + (-4.94 + 15.2i)T^{2} \) |
| 5 | \( 1 + (32.0 + 23.2i)T + (193. + 594. i)T^{2} \) |
| 7 | \( 1 + (-27.3 - 8.87i)T + (1.94e3 + 1.41e3i)T^{2} \) |
| 13 | \( 1 + (-94.6 - 130. i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (141. - 195. i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (-439. + 142. i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 + 241.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-553. - 179. i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (-580. + 421. i)T + (2.85e5 - 8.78e5i)T^{2} \) |
| 37 | \( 1 + (-757. + 2.33e3i)T + (-1.51e6 - 1.10e6i)T^{2} \) |
| 41 | \( 1 + (-529. + 172. i)T + (2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 - 3.16e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-777. - 2.39e3i)T + (-3.94e6 + 2.86e6i)T^{2} \) |
| 53 | \( 1 + (-2.71e3 + 1.97e3i)T + (2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (850. - 2.61e3i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + (3.69e3 - 5.08e3i)T + (-4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 - 309.T + 2.01e7T^{2} \) |
| 71 | \( 1 + (4.76e3 + 3.46e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (2.84e3 + 923. i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (2.71e3 + 3.73e3i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (4.05e3 - 5.58e3i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 - 6.52e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-5.07e3 + 3.69e3i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−15.87934400865683107997036364507, −14.51417367972907134031706510560, −13.36484859987824635621516015973, −11.82241430769892416085239807828, −11.12029759017930659632346324121, −8.875723445020161187403099148930, −7.64229527884486113332889862231, −6.00651601053758149470021404563, −4.44153001088637574544305492792, −1.04248721126083193198288683330,
3.18632078630861746469983171388, 4.46432870592376678894658199071, 7.11399508949871954293759246724, 8.151565841054381891093381578207, 10.33617653925648589579104956714, 11.46771504880559481479550454356, 12.08281131160099534920585442841, 13.88396231504442304747031587271, 15.27645553949264086992744620832, 15.93641946090259956075104170550
