| L(s) = 1 | + (1.85 + 3.22i)2-s + (2.47 + 1.69i)3-s + (−4.91 + 8.51i)4-s + (0.0731 − 4.99i)5-s + (−0.869 + 11.1i)6-s + (4.87 − 5.02i)7-s − 21.7·8-s + (3.23 + 8.39i)9-s + (16.2 − 9.06i)10-s + (−10.0 − 5.82i)11-s + (−26.6 + 12.7i)12-s − 9.22i·13-s + (25.2 + 6.37i)14-s + (8.66 − 12.2i)15-s + (−20.7 − 35.8i)16-s + (1.56 − 2.70i)17-s + ⋯ |
| L(s) = 1 | + (0.929 + 1.61i)2-s + (0.824 + 0.565i)3-s + (−1.22 + 2.12i)4-s + (0.0146 − 0.999i)5-s + (−0.144 + 1.85i)6-s + (0.696 − 0.717i)7-s − 2.71·8-s + (0.359 + 0.933i)9-s + (1.62 − 0.906i)10-s + (−0.916 − 0.529i)11-s + (−2.21 + 1.05i)12-s − 0.709i·13-s + (1.80 + 0.455i)14-s + (0.577 − 0.816i)15-s + (−1.29 − 2.24i)16-s + (0.0919 − 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09805 + 2.19755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09805 + 2.19755i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.47 - 1.69i)T \) |
| 5 | \( 1 + (-0.0731 + 4.99i)T \) |
| 7 | \( 1 + (-4.87 + 5.02i)T \) |
| good | 2 | \( 1 + (-1.85 - 3.22i)T + (-2 + 3.46i)T^{2} \) |
| 11 | \( 1 + (10.0 + 5.82i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 9.22iT - 169T^{2} \) |
| 17 | \( 1 + (-1.56 + 2.70i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-5.39 - 9.34i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (2.93 + 5.08i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 38.3iT - 841T^{2} \) |
| 31 | \( 1 + (15.7 - 27.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-20.0 + 11.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 22.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 29.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-30.0 - 52.1i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-27.9 + 48.4i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (23.2 + 13.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (19.8 + 34.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (86.4 + 49.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 62.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (37.5 + 21.6i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-15.5 - 26.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 93.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-34.2 + 19.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 119. iT - 9.40e3T^{2} \) |
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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
−14.04172988737080627073791737436, −13.33833283318940151895000733207, −12.48439195279204202660360070408, −10.57857904312976908462364605502, −8.961649915861277914935158017245, −8.112354202190686034560607041285, −7.46596934589758964441569440907, −5.48882445111964961318519954134, −4.77610854984540169948831086863, −3.54039004049401842934419503680,
2.01297505667700787949959751036, 2.82044336280515490188121979963, 4.32510739936400039337854519533, 5.94186724913587159804311104111, 7.64844537172061327516288769688, 9.219370263135704144865856189203, 10.19424344865568012177626637381, 11.39575019743583979657354714635, 12.02943192538029651454776574426, 13.22042727941601699504347305052
