| 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
−15.93641946090259956075104170550, −15.27645553949264086992744620832, −13.88396231504442304747031587271, −12.08281131160099534920585442841, −11.46771504880559481479550454356, −10.33617653925648589579104956714, −8.151565841054381891093381578207, −7.11399508949871954293759246724, −4.46432870592376678894658199071, −3.18632078630861746469983171388,
1.04248721126083193198288683330, 4.44153001088637574544305492792, 6.00651601053758149470021404563, 7.64229527884486113332889862231, 8.875723445020161187403099148930, 11.12029759017930659632346324121, 11.82241430769892416085239807828, 13.36484859987824635621516015973, 14.51417367972907134031706510560, 15.87934400865683107997036364507
