| L(s) = 1 | + (−5.20 − 10.8i)2-s + (3.59 − 0.269i)3-s + (−49.8 + 62.5i)4-s + (−53.9 − 174. i)5-s + (−21.6 − 37.4i)6-s + (85.2 + 49.2i)7-s + (187. + 42.8i)8-s + (−708. + 106. i)9-s + (−1.60e3 + 1.49e3i)10-s + (−88.2 − 110. i)11-s + (−162. + 238. i)12-s + (46.5 + 43.2i)13-s + (88.3 − 1.17e3i)14-s + (−240. − 613. i)15-s + (625. + 2.74e3i)16-s + (4.92e3 + 1.52e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.650 − 1.35i)2-s + (0.133 − 0.00997i)3-s + (−0.779 + 0.977i)4-s + (−0.431 − 1.39i)5-s + (−0.100 − 0.173i)6-s + (0.248 + 0.143i)7-s + (0.366 + 0.0836i)8-s + (−0.971 + 0.146i)9-s + (−1.60 + 1.49i)10-s + (−0.0662 − 0.0830i)11-s + (−0.0940 + 0.137i)12-s + (0.0212 + 0.0196i)13-s + (0.0321 − 0.429i)14-s + (−0.0713 − 0.181i)15-s + (0.152 + 0.669i)16-s + (1.00 + 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.209983 + 0.186457i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.209983 + 0.186457i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.36e4 + 2.98e4i)T \) |
| good | 2 | \( 1 + (5.20 + 10.8i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-3.59 + 0.269i)T + (720. - 108. i)T^{2} \) |
| 5 | \( 1 + (53.9 + 174. i)T + (-1.29e4 + 8.80e3i)T^{2} \) |
| 7 | \( 1 + (-85.2 - 49.2i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (88.2 + 110. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-46.5 - 43.2i)T + (3.60e5 + 4.81e6i)T^{2} \) |
| 17 | \( 1 + (-4.92e3 - 1.52e3i)T + (1.99e7 + 1.35e7i)T^{2} \) |
| 19 | \( 1 + (571. - 3.78e3i)T + (-4.49e7 - 1.38e7i)T^{2} \) |
| 23 | \( 1 + (-629. + 1.60e3i)T + (-1.08e8 - 1.00e8i)T^{2} \) |
| 29 | \( 1 + (2.40e4 + 1.80e3i)T + (5.88e8 + 8.86e7i)T^{2} \) |
| 31 | \( 1 + (1.11e4 + 7.57e3i)T + (3.24e8 + 8.26e8i)T^{2} \) |
| 37 | \( 1 + (6.78e4 - 3.91e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (5.97e4 - 2.87e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-7.27e4 + 9.11e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-8.78e4 + 8.15e4i)T + (1.65e9 - 2.21e10i)T^{2} \) |
| 59 | \( 1 + (6.51e4 + 2.85e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.05e5 - 1.54e5i)T + (-1.88e10 + 4.79e10i)T^{2} \) |
| 67 | \( 1 + (9.87e4 + 1.48e4i)T + (8.64e10 + 2.66e10i)T^{2} \) |
| 71 | \( 1 + (4.70e5 - 1.84e5i)T + (9.39e10 - 8.71e10i)T^{2} \) |
| 73 | \( 1 + (2.43e5 - 2.62e5i)T + (-1.13e10 - 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-3.83e5 + 6.63e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-1.75e4 - 2.34e5i)T + (-3.23e11 + 4.87e10i)T^{2} \) |
| 89 | \( 1 + (4.25e5 - 3.19e4i)T + (4.91e11 - 7.40e10i)T^{2} \) |
| 97 | \( 1 + (1.04e5 + 1.31e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−13.30357613455636145041549243400, −12.19528932206408099788900309624, −11.55175623350278303513951130140, −10.12870045096653268689503212083, −8.817437231399615571239626875053, −8.194342522566159386068800891969, −5.38158333518705497025704795975, −3.55084577318836630475000125302, −1.65994220642251470876765802682, −0.15995203048231139584666233221,
3.13313257150685819938226166850, 5.62731361577570917845657699410, 6.95489158025440967277076862491, 7.79212742764380279303873572529, 9.099685484690475052053887134987, 10.60357804875374140005282179632, 11.80549630111299460476796436990, 14.01297083357496156819154983275, 14.70188196992136218788655883944, 15.46644749113382739058283401967
