| L(s) = 1 | + (−2.24 − 1.79i)2-s + (0.540 + 3.58i)3-s + (0.948 + 4.15i)4-s + (3.03 + 4.44i)5-s + (5.21 − 9.02i)6-s + (5.12 − 2.95i)7-s + (0.324 − 0.674i)8-s + (−3.97 + 1.22i)9-s + (1.15 − 15.4i)10-s + (−1.83 + 8.05i)11-s + (−14.3 + 5.64i)12-s + (0.572 + 7.63i)13-s + (−16.8 − 2.53i)14-s + (−14.3 + 13.2i)15-s + (13.4 − 6.46i)16-s + (−27.9 − 19.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.12 − 0.895i)2-s + (0.180 + 1.19i)3-s + (0.237 + 1.03i)4-s + (0.606 + 0.889i)5-s + (0.868 − 1.50i)6-s + (0.731 − 0.422i)7-s + (0.0405 − 0.0842i)8-s + (−0.441 + 0.136i)9-s + (0.115 − 1.54i)10-s + (−0.167 + 0.731i)11-s + (−1.19 + 0.470i)12-s + (0.0440 + 0.587i)13-s + (−1.20 − 0.180i)14-s + (−0.953 + 0.885i)15-s + (0.838 − 0.403i)16-s + (−1.64 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.734301 + 0.157122i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734301 + 0.157122i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-30.2 - 30.5i)T \) |
| good | 2 | \( 1 + (2.24 + 1.79i)T + (0.890 + 3.89i)T^{2} \) |
| 3 | \( 1 + (-0.540 - 3.58i)T + (-8.60 + 2.65i)T^{2} \) |
| 5 | \( 1 + (-3.03 - 4.44i)T + (-9.13 + 23.2i)T^{2} \) |
| 7 | \( 1 + (-5.12 + 2.95i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (1.83 - 8.05i)T + (-109. - 52.4i)T^{2} \) |
| 13 | \( 1 + (-0.572 - 7.63i)T + (-167. + 25.1i)T^{2} \) |
| 17 | \( 1 + (27.9 + 19.0i)T + (105. + 269. i)T^{2} \) |
| 19 | \( 1 + (-7.85 + 25.4i)T + (-298. - 203. i)T^{2} \) |
| 23 | \( 1 + (-19.6 - 18.2i)T + (39.5 + 527. i)T^{2} \) |
| 29 | \( 1 + (-6.81 + 45.2i)T + (-803. - 247. i)T^{2} \) |
| 31 | \( 1 + (5.75 + 14.6i)T + (-704. + 653. i)T^{2} \) |
| 37 | \( 1 + (28.6 + 16.5i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-4.06 + 5.09i)T + (-374. - 1.63e3i)T^{2} \) |
| 47 | \( 1 + (-6.13 - 26.8i)T + (-1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (0.330 - 4.40i)T + (-2.77e3 - 418. i)T^{2} \) |
| 59 | \( 1 + (-73.2 + 35.2i)T + (2.17e3 - 2.72e3i)T^{2} \) |
| 61 | \( 1 + (1.94 + 0.764i)T + (2.72e3 + 2.53e3i)T^{2} \) |
| 67 | \( 1 + (-0.292 - 0.0903i)T + (3.70e3 + 2.52e3i)T^{2} \) |
| 71 | \( 1 + (61.0 + 65.7i)T + (-376. + 5.02e3i)T^{2} \) |
| 73 | \( 1 + (48.3 - 3.62i)T + (5.26e3 - 794. i)T^{2} \) |
| 79 | \( 1 + (12.9 + 22.3i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (0.0352 - 0.00531i)T + (6.58e3 - 2.03e3i)T^{2} \) |
| 89 | \( 1 + (2.64 + 17.5i)T + (-7.56e3 + 2.33e3i)T^{2} \) |
| 97 | \( 1 + (14.4 - 63.2i)T + (-8.47e3 - 4.08e3i)T^{2} \) |
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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
−15.83830039028204566712171313654, −14.81923713444386702156425766847, −13.62544426708774507324295626778, −11.38636366787365110033513108222, −10.86648303942970704189033450184, −9.721963419034913128750553290570, −9.110224954175108795478523607664, −7.19564170719915037812927170990, −4.65201549792694774918318862582, −2.50368606717622979999470353622,
1.43542392069503288444141689020, 5.60923692172926549698594433977, 6.92611558662070598450249498751, 8.419187144878559359726851702723, 8.673768710650789430812767638036, 10.54241667770225991998143554688, 12.45216040027015973940559772552, 13.23622016158160599707089231831, 14.66228881222569718300332763106, 15.99430560665608149198129074887
