L(s) = 1 | + (0.965 + 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s − i·6-s + (−2.63 − 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−2.54 + 4.40i)11-s + (0.258 − 0.965i)12-s + (−2.02 + 2.02i)13-s + (−2.47 − 0.931i)14-s + (0.500 + 0.866i)16-s + (−6.71 + 1.79i)17-s + (−0.965 + 0.258i)18-s + (−1.79 − 3.11i)19-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s − 0.408i·6-s + (−0.995 − 0.0978i)7-s + (0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.767 + 1.32i)11-s + (0.0747 − 0.278i)12-s + (−0.561 + 0.561i)13-s + (−0.661 − 0.248i)14-s + (0.125 + 0.216i)16-s + (−1.62 + 0.436i)17-s + (−0.227 + 0.0610i)18-s + (−0.412 − 0.714i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6109033030\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6109033030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.63 + 0.258i)T \) |
good | 11 | \( 1 + (2.54 - 4.40i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.02 - 2.02i)T - 13iT^{2} \) |
| 17 | \( 1 + (6.71 - 1.79i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.79 + 3.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.79 + 6.71i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 8.81iT - 29T^{2} \) |
| 31 | \( 1 + (-1.61 - 0.931i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.50 + 2.01i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 4.13iT - 41T^{2} \) |
| 43 | \( 1 + (-7.84 - 7.84i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.482 - 1.79i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.879 - 0.235i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (2.08 - 3.61i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.08 - 2.93i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.762 + 2.84i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 7.86T + 71T^{2} \) |
| 73 | \( 1 + (3.50 + 13.0i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (10.4 - 6.04i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.99 + 1.99i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.82 - 6.62i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.4 + 13.4i)T + 97iT^{2} \) |
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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
−10.51097057107083507869171326873, −9.322643227650508620262455919762, −8.593315029224634634158117550983, −7.23139738323793772986083891315, −6.93712803720138176796043001729, −6.19502841959877572184255974925, −4.90940716856902714528228479221, −4.34547453714575308475911805643, −2.84762141282124378179167312787, −2.07532545694154417790593529514,
0.19830459316095912694763586583, 2.46255744477638733448317946906, 3.28455976861678910797693473528, 4.18690398451688921884357983339, 5.34172520465401613057796126660, 5.91396775987100844168142813684, 6.80558783170711416357606979962, 7.919256329831398668754221886465, 8.903426433593636262988048352235, 9.757671717442153072779353695801