| L(s) = 1 | + (4.47 + 9.29i)2-s + (53.8 + 78.9i)3-s + (93.3 − 117. i)4-s + (−87.0 + 93.7i)5-s + (−492. + 853. i)6-s + (2.15e3 − 1.24e3i)7-s + (4.07e3 + 930. i)8-s + (−940. + 2.39e3i)9-s + (−1.26e3 − 388. i)10-s + (1.19e4 + 1.49e4i)11-s + (1.42e4 + 1.06e3i)12-s + (2.24e3 − 691. i)13-s + (2.11e4 + 1.44e4i)14-s + (−1.20e4 − 1.82e3i)15-s + (1.07e3 + 4.70e3i)16-s + (8.88e4 − 8.24e4i)17-s + ⋯ |
| L(s) = 1 | + (0.279 + 0.580i)2-s + (0.664 + 0.974i)3-s + (0.364 − 0.457i)4-s + (−0.139 + 0.150i)5-s + (−0.380 + 0.658i)6-s + (0.896 − 0.517i)7-s + (0.995 + 0.227i)8-s + (−0.143 + 0.365i)9-s + (−0.126 − 0.0388i)10-s + (0.812 + 1.01i)11-s + (0.687 + 0.0515i)12-s + (0.0784 − 0.0242i)13-s + (0.551 + 0.375i)14-s + (−0.238 − 0.0360i)15-s + (0.0163 + 0.0717i)16-s + (1.06 − 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.74567 + 2.08667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74567 + 2.08667i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.77e5 + 3.39e6i)T \) |
| good | 2 | \( 1 + (-4.47 - 9.29i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (-53.8 - 78.9i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (87.0 - 93.7i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-2.15e3 + 1.24e3i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-1.19e4 - 1.49e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-2.24e3 + 691. i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (-8.88e4 + 8.24e4i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (1.96e5 - 7.69e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (3.75e5 - 5.65e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-1.31e5 + 1.93e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (7.32e4 - 9.76e5i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (-3.70e5 - 2.13e5i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (3.90e6 - 1.88e6i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-4.53e6 + 5.68e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (9.20e6 + 2.84e6i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (4.15e6 + 1.81e7i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-1.67e7 + 1.25e6i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (-1.22e6 - 3.13e6i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (2.47e5 - 1.64e6i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (5.73e6 + 1.85e7i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (2.21e5 + 3.83e5i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-3.94e7 + 2.68e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (-3.44e7 - 5.05e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (4.91e7 + 6.16e7i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−14.61257537922454377276471467821, −14.01909234372216263417940140167, −11.96522210581616660490306489868, −10.56986770281202774952183369258, −9.702845257612614883007947176169, −8.076207554813682603392983094193, −6.80169836070864679898715493342, −4.99741470370860160945876311033, −3.89359090533365043431558563422, −1.69360939649973416251387395074,
1.39272171643087713204032142190, 2.46287064821091959258242020807, 4.07678158895929603252072573004, 6.31479562178267019619026736174, 7.952257916558415871408988703611, 8.488313497636491879069510358481, 10.71827554846646535451021261222, 11.88087590785107915292071191658, 12.64684187843398381919570903339, 13.79235348719397300222060336269
