| L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (0.247 − 0.294i)5-s + (−2.50 − 2.09i)7-s + (−0.866 + 0.5i)8-s + (−0.192 + 0.333i)10-s + (1.29 + 2.23i)11-s + (−0.466 − 1.28i)13-s + (2.82 + 1.63i)14-s + (0.766 − 0.642i)16-s + (1.13 − 3.13i)17-s + (−5.89 − 1.03i)19-s + (0.131 − 0.361i)20-s + (−1.66 − 1.98i)22-s + (−6.53 − 3.77i)23-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.469 − 0.171i)4-s + (0.110 − 0.131i)5-s + (−0.945 − 0.793i)7-s + (−0.306 + 0.176i)8-s + (−0.0608 + 0.105i)10-s + (0.389 + 0.674i)11-s + (−0.129 − 0.355i)13-s + (0.755 + 0.436i)14-s + (0.191 − 0.160i)16-s + (0.276 − 0.759i)17-s + (−1.35 − 0.238i)19-s + (0.0294 − 0.0808i)20-s + (−0.354 − 0.422i)22-s + (−1.36 − 0.786i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.136688 - 0.380479i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.136688 - 0.380479i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-0.543 + 6.05i)T \) |
| good | 5 | \( 1 + (-0.247 + 0.294i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.50 + 2.09i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-1.29 - 2.23i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.466 + 1.28i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.13 + 3.13i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (5.89 + 1.03i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (6.53 + 3.77i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.78 - 1.60i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.53iT - 31T^{2} \) |
| 41 | \( 1 + (7.77 - 2.82i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 - 4.33iT - 43T^{2} \) |
| 47 | \( 1 + (-2.61 + 4.52i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.64 + 5.57i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (8.33 + 9.93i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.39 + 6.58i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (8.60 + 7.22i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.60 - 14.7i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + (-0.940 + 1.12i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.0104 + 0.00380i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.612 - 0.730i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-12.8 - 7.40i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.03833362879039776503982747594, −9.466393202715480604099469933275, −8.523014295225440856566956149062, −7.46419126543612011807708835003, −6.81194332456490766913817735883, −5.93923304243480058848893629135, −4.57338916692161725576830802087, −3.43684369406898987111630978775, −2.00626002608542527129342900166, −0.25389976728869245259753105109,
1.84779077836505118556444181594, 3.04664155857620972791245791439, 4.15418958640919460799055541564, 6.03693246569663202223498527866, 6.15153483007323053819535857062, 7.44481206551226673142329743857, 8.579177173110542916355767760859, 8.973177352374044066003950988477, 10.09871479281111193400956306863, 10.51001923864513220039304290337
