L(s) = 1 | + (−1.41 − 0.0874i)2-s + (0.0727 + 1.73i)3-s + (1.98 + 0.246i)4-s + (−0.723 + 0.0633i)5-s + (0.0486 − 2.44i)6-s + (0.398 − 1.09i)7-s + (−2.77 − 0.522i)8-s + (−2.98 + 0.251i)9-s + (1.02 − 0.0260i)10-s + (−0.304 + 3.48i)11-s + (−0.282 + 3.45i)12-s + (−0.440 + 0.308i)13-s + (−0.658 + 1.51i)14-s + (−0.162 − 1.24i)15-s + (3.87 + 0.980i)16-s + (−3.29 + 5.70i)17-s + ⋯ |
L(s) = 1 | + (−0.998 − 0.0618i)2-s + (0.0420 + 0.999i)3-s + (0.992 + 0.123i)4-s + (−0.323 + 0.0283i)5-s + (0.0198 − 0.999i)6-s + (0.150 − 0.414i)7-s + (−0.982 − 0.184i)8-s + (−0.996 + 0.0839i)9-s + (0.324 − 0.00824i)10-s + (−0.0918 + 1.04i)11-s + (−0.0816 + 0.996i)12-s + (−0.122 + 0.0854i)13-s + (−0.176 + 0.404i)14-s + (−0.0418 − 0.322i)15-s + (0.969 + 0.245i)16-s + (−0.799 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0881675 + 0.472764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0881675 + 0.472764i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 0.0874i)T \) |
| 3 | \( 1 + (-0.0727 - 1.73i)T \) |
good | 5 | \( 1 + (0.723 - 0.0633i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-0.398 + 1.09i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.304 - 3.48i)T + (-10.8 - 1.91i)T^{2} \) |
| 13 | \( 1 + (0.440 - 0.308i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (3.29 - 5.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0169 + 0.0632i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.415 + 1.14i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.23 + 1.76i)T + (-9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (3.93 - 1.43i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.92 - 7.20i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (6.52 + 1.15i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.375 + 4.28i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (6.70 + 2.43i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (3.80 - 3.80i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.51 + 0.482i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-1.21 - 2.59i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (3.74 - 2.62i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-9.89 - 5.71i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (9.82 - 5.67i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.701 + 3.97i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-8.08 + 11.5i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (1.53 - 0.885i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.49 - 2.93i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.27415840439386470354171445447, −10.36268654890785262939260070874, −9.946613662755526571644200504092, −8.874473248771204860958882345118, −8.144775936899597211045093055708, −7.13939722531334341660785257258, −6.03394027713201759033604260828, −4.60579665690961142783951337605, −3.59046803429312668395957165715, −2.04578548150993134070864776662,
0.39153953722588044191981941695, 2.07074388600098883231437291737, 3.20890784455887364325088648281, 5.35819637941527261359634225835, 6.31646483095435432509322878272, 7.24512887025545339443627591157, 8.042156195565090968455584951102, 8.776482116511504378232278005097, 9.593367069498333386507205942192, 11.03818522016658226576866395830