L(s) = 1 | + (−0.351 − 2.44i)2-s + (−0.909 − 0.415i)3-s + (−3.92 + 1.15i)4-s + (−2.70 + 3.12i)5-s + (−0.695 + 2.36i)6-s + (−0.374 − 2.61i)7-s + (2.13 + 4.68i)8-s + (0.654 + 0.755i)9-s + (8.57 + 5.51i)10-s + (2.25 + 0.324i)11-s + (4.04 + 0.581i)12-s + (−0.473 + 0.736i)13-s + (−6.26 + 1.83i)14-s + (3.75 − 1.71i)15-s + (3.81 − 2.44i)16-s + (4.80 + 1.41i)17-s + ⋯ |
L(s) = 1 | + (−0.248 − 1.72i)2-s + (−0.525 − 0.239i)3-s + (−1.96 + 0.575i)4-s + (−1.20 + 1.39i)5-s + (−0.283 + 0.966i)6-s + (−0.141 − 0.989i)7-s + (0.756 + 1.65i)8-s + (0.218 + 0.251i)9-s + (2.71 + 1.74i)10-s + (0.681 + 0.0979i)11-s + (1.16 + 0.167i)12-s + (−0.131 + 0.204i)13-s + (−1.67 + 0.489i)14-s + (0.970 − 0.443i)15-s + (0.952 − 0.612i)16-s + (1.16 + 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.560777 - 0.285347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.560777 - 0.285347i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (0.374 + 2.61i)T \) |
| 23 | \( 1 + (-4.44 - 1.79i)T \) |
good | 2 | \( 1 + (0.351 + 2.44i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (2.70 - 3.12i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.25 - 0.324i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.473 - 0.736i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.80 - 1.41i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (3.12 - 0.918i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.842 + 0.247i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-6.39 + 2.91i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.73 - 4.97i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.97 - 1.70i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-7.21 - 3.29i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 7.08iT - 47T^{2} \) |
| 53 | \( 1 + (-4.26 - 6.63i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (3.91 - 6.08i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.502 - 1.10i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-10.3 + 1.49i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.86 - 12.9i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.659 - 2.24i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-5.50 + 8.56i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (1.88 + 2.17i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.95 + 13.0i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (6.42 - 7.41i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.82093500488610170717093814994, −10.46030346576726047454889556324, −9.572864094136428251806228426099, −8.151519823255088680572606392097, −7.31947862981508256312466982978, −6.39880067716805268461026955842, −4.41493553394905622947833029671, −3.74959878912783843334788784272, −2.82482006551378781592528680027, −1.06793532814473239580124955539,
0.59455667170823852859345323877, 3.82713540296423182281571550479, 4.96747135300750543294986826218, 5.37468543750979404469438186512, 6.51958329851713560444364625164, 7.49799509579781199652733374545, 8.470116602165200299983267019277, 8.856548315148958249800882868252, 9.702925943309134643723047850439, 11.25004605367051044436601046923