| L(s) = 1 | + (0.170 − 0.469i)2-s + (1.34 + 1.12i)4-s + (0.000430 + 0.000361i)5-s + (1.75 − 1.98i)7-s + (1.62 − 0.936i)8-s + (0.000243 − 0.000140i)10-s + (−3.04 − 3.62i)11-s + (1.77 − 2.10i)13-s + (−0.632 − 1.16i)14-s + (0.445 + 2.52i)16-s + (0.400 + 0.692i)17-s + (4.50 + 2.60i)19-s + (0.000170 + 0.000969i)20-s + (−2.22 + 0.808i)22-s + (−0.542 − 1.49i)23-s + ⋯ |
| L(s) = 1 | + (0.120 − 0.331i)2-s + (0.670 + 0.562i)4-s + (0.000192 + 0.000161i)5-s + (0.661 − 0.749i)7-s + (0.573 − 0.331i)8-s + (7.68e−5 − 4.43e−5i)10-s + (−0.917 − 1.09i)11-s + (0.490 − 0.585i)13-s + (−0.168 − 0.310i)14-s + (0.111 + 0.631i)16-s + (0.0970 + 0.168i)17-s + (1.03 + 0.597i)19-s + (3.82e−5 + 0.000216i)20-s + (−0.473 + 0.172i)22-s + (−0.113 − 0.310i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.86517 - 0.596768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86517 - 0.596768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.75 + 1.98i)T \) |
| good | 2 | \( 1 + (-0.170 + 0.469i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.000430 - 0.000361i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (3.04 + 3.62i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.77 + 2.10i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.400 - 0.692i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.50 - 2.60i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.542 + 1.49i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-5.40 - 6.44i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (5.19 - 6.18i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 - 2.05T + 37T^{2} \) |
| 41 | \( 1 + (-5.20 - 4.36i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.39 + 2.32i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-5.68 + 4.76i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (9.11 + 5.26i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.29 - 13.0i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.33 + 2.78i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.99 - 0.727i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (7.56 + 4.37i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 7.18iT - 73T^{2} \) |
| 79 | \( 1 + (2.42 + 0.881i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.47 - 1.23i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.61 + 2.80i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.614 + 1.68i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.59526528659115376587757817264, −10.33705662637309346469305409825, −8.595796842661874541211278994958, −8.004595977451416507159399330224, −7.24078789607781765527414900998, −6.09518841350340478875670851176, −4.99494114696580989655516798692, −3.68463478050456295196223641422, −2.88037582682140506389573222925, −1.26881512390512282569415770544,
1.67939343024668676504444581968, 2.69265244327573992162232696399, 4.53992578088505567947714249800, 5.35713262002035124346732409098, 6.16657525831804170332154840662, 7.38624814832237875612092957055, 7.83049915237275231152393739840, 9.223200773610279776610262318439, 9.879202194021273089087080266186, 11.04253701506920961337800587797
