| L(s) = 1 | + (3.91 − 8.12i)2-s + (−30.1 + 44.2i)3-s + (−10.7 − 13.4i)4-s + (−104. − 112. i)5-s + (241. + 417. i)6-s + (148. + 85.7i)7-s + (410. − 93.7i)8-s + (−780. − 1.98e3i)9-s + (−1.32e3 + 407. i)10-s + (1.37e3 − 1.72e3i)11-s + (920. − 68.9i)12-s + (−1.26e3 − 388. i)13-s + (1.27e3 − 871. i)14-s + (8.12e3 − 1.22e3i)15-s + (1.09e3 − 4.77e3i)16-s + (3.90e3 + 3.62e3i)17-s + ⋯ |
| L(s) = 1 | + (0.488 − 1.01i)2-s + (−1.11 + 1.63i)3-s + (−0.168 − 0.210i)4-s + (−0.835 − 0.900i)5-s + (1.11 + 1.93i)6-s + (0.433 + 0.250i)7-s + (0.802 − 0.183i)8-s + (−1.07 − 2.72i)9-s + (−1.32 + 0.407i)10-s + (1.03 − 1.29i)11-s + (0.532 − 0.0399i)12-s + (−0.573 − 0.176i)13-s + (0.465 − 0.317i)14-s + (2.40 − 0.362i)15-s + (0.266 − 1.16i)16-s + (0.794 + 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.735695 - 0.904140i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.735695 - 0.904140i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.83e4 + 7.42e4i)T \) |
| good | 2 | \( 1 + (-3.91 + 8.12i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (30.1 - 44.2i)T + (-266. - 678. i)T^{2} \) |
| 5 | \( 1 + (104. + 112. i)T + (-1.16e3 + 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-148. - 85.7i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.37e3 + 1.72e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (1.26e3 + 388. i)T + (3.98e6 + 2.71e6i)T^{2} \) |
| 17 | \( 1 + (-3.90e3 - 3.62e3i)T + (1.80e6 + 2.40e7i)T^{2} \) |
| 19 | \( 1 + (5.12e3 + 2.01e3i)T + (3.44e7 + 3.19e7i)T^{2} \) |
| 23 | \( 1 + (7.17e3 + 1.08e3i)T + (1.41e8 + 4.36e7i)T^{2} \) |
| 29 | \( 1 + (1.79e4 + 2.63e4i)T + (-2.17e8 + 5.53e8i)T^{2} \) |
| 31 | \( 1 + (1.75e3 + 2.34e4i)T + (-8.77e8 + 1.32e8i)T^{2} \) |
| 37 | \( 1 + (2.69e3 - 1.55e3i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-3.00e4 - 1.44e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (5.74e3 + 7.21e3i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (5.53e4 - 1.70e4i)T + (1.83e10 - 1.24e10i)T^{2} \) |
| 59 | \( 1 + (-2.42e4 + 1.06e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (2.70e4 + 2.02e3i)T + (5.09e10 + 7.67e9i)T^{2} \) |
| 67 | \( 1 + (4.23e4 - 1.08e5i)T + (-6.63e10 - 6.15e10i)T^{2} \) |
| 71 | \( 1 + (-5.63e4 - 3.73e5i)T + (-1.22e11 + 3.77e10i)T^{2} \) |
| 73 | \( 1 + (1.20e5 - 3.91e5i)T + (-1.25e11 - 8.52e10i)T^{2} \) |
| 79 | \( 1 + (1.80e5 - 3.12e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.33e5 + 9.09e4i)T + (1.19e11 + 3.04e11i)T^{2} \) |
| 89 | \( 1 + (1.72e4 - 2.53e4i)T + (-1.81e11 - 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-2.28e5 + 2.86e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−14.51740221879392532401554046197, −12.52138108555318712526383617727, −11.67474453640967842616256864828, −11.17564468963140727274216055007, −9.903099944146630269713217520876, −8.500447045092051024184273522679, −5.75841818000081956047439484890, −4.40125152981718494797893860113, −3.70609813290368179965956827791, −0.56727407043149025469344962577,
1.65095514990959203484062208468, 4.79692817785575776434997597162, 6.30793540974141529924919501988, 7.25171735109964486722236763703, 7.61667882989888185347098137519, 10.70300282583680071899365054608, 11.72970650258326211558302497239, 12.60244287228929787472518441944, 14.17178348890330594749960238511, 14.70650721045296161078429827537
