| L(s) = 1 | + (0.00284 − 0.0325i)2-s + (2.50 + 1.64i)3-s + (3.93 + 0.694i)4-s + (−4.23 + 2.65i)5-s + (0.0608 − 0.0768i)6-s + (3.26 + 2.28i)7-s + (0.0676 − 0.252i)8-s + (3.56 + 8.26i)9-s + (0.0744 + 0.145i)10-s + (−7.95 − 2.89i)11-s + (8.72 + 8.23i)12-s + (−0.859 − 9.81i)13-s + (0.0837 − 0.0997i)14-s + (−14.9 − 0.322i)15-s + (15.0 + 5.46i)16-s + (−0.380 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (0.00142 − 0.0162i)2-s + (0.835 + 0.549i)3-s + (0.984 + 0.173i)4-s + (−0.846 + 0.531i)5-s + (0.0101 − 0.0128i)6-s + (0.466 + 0.326i)7-s + (0.00845 − 0.0315i)8-s + (0.395 + 0.918i)9-s + (0.00744 + 0.0145i)10-s + (−0.722 − 0.263i)11-s + (0.727 + 0.686i)12-s + (−0.0660 − 0.755i)13-s + (0.00597 − 0.00712i)14-s + (−0.999 − 0.0214i)15-s + (0.938 + 0.341i)16-s + (−0.0223 − 0.0835i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.76879 + 0.895402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76879 + 0.895402i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.50 - 1.64i)T \) |
| 5 | \( 1 + (4.23 - 2.65i)T \) |
| good | 2 | \( 1 + (-0.00284 + 0.0325i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-3.26 - 2.28i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (7.95 + 2.89i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.859 + 9.81i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (0.380 + 1.41i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (4.51 - 2.60i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-28.2 + 19.7i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-12.4 - 14.7i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (1.18 - 6.73i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (14.5 + 54.3i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (39.7 + 33.3i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-2.36 - 5.07i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (33.5 + 23.4i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (28.5 + 28.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (3.35 + 9.22i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-19.4 - 110. i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (33.5 - 2.93i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (42.8 - 74.2i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-26.2 + 97.9i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-21.3 - 25.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-147. - 12.9i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-31.1 + 17.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (134. - 62.7i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−13.09396081394440428431610398745, −12.03363345239417105366046049056, −10.82476519367676626938979040063, −10.46773503968804116278119893987, −8.661017823003739321821512903532, −7.897642484912665986452510362324, −6.94437946173451204457557924553, −5.15224629162151517508550670718, −3.48468217484637770574332910122, −2.49884462864137014941402917811,
1.51259679208473795649900087686, 3.13828214269655186088189528135, 4.75803304512375537759685988726, 6.63840920210893009067502567057, 7.56843539757434742831995119265, 8.290938134496487685234742743040, 9.621371920208652452706931073603, 11.03790355638999765410806853099, 11.85115798799993451065999086137, 12.79898812706049922528176120828
