| L(s) = 1 | + (5.75 − 11.9i)2-s + (−1.56 − 0.117i)3-s + (−69.8 − 87.5i)4-s + (36.5 − 118. i)5-s + (−10.3 + 17.9i)6-s + (348. − 200. i)7-s + (−620. + 141. i)8-s + (−718. − 108. i)9-s + (−1.20e3 − 1.12e3i)10-s + (−1.06e3 + 1.33e3i)11-s + (98.8 + 144. i)12-s + (−1.11e3 + 1.03e3i)13-s + (−398. − 5.31e3i)14-s + (−71.0 + 181. i)15-s + (−284. + 1.24e3i)16-s + (7.37e3 − 2.27e3i)17-s + ⋯ |
| L(s) = 1 | + (0.719 − 1.49i)2-s + (−0.0578 − 0.00433i)3-s + (−1.09 − 1.36i)4-s + (0.292 − 0.948i)5-s + (−0.0481 + 0.0833i)6-s + (1.01 − 0.585i)7-s + (−1.21 + 0.276i)8-s + (−0.985 − 0.148i)9-s + (−1.20 − 1.12i)10-s + (−0.799 + 1.00i)11-s + (0.0571 + 0.0838i)12-s + (−0.508 + 0.471i)13-s + (−0.145 − 1.93i)14-s + (−0.0210 + 0.0536i)15-s + (−0.0693 + 0.303i)16-s + (1.50 − 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0983226 + 2.28077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0983226 + 2.28077i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.42e4 + 7.82e4i)T \) |
| good | 2 | \( 1 + (-5.75 + 11.9i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (1.56 + 0.117i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (-36.5 + 118. i)T + (-1.29e4 - 8.80e3i)T^{2} \) |
| 7 | \( 1 + (-348. + 200. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.06e3 - 1.33e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (1.11e3 - 1.03e3i)T + (3.60e5 - 4.81e6i)T^{2} \) |
| 17 | \( 1 + (-7.37e3 + 2.27e3i)T + (1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (-368. - 2.44e3i)T + (-4.49e7 + 1.38e7i)T^{2} \) |
| 23 | \( 1 + (7.07e3 + 1.80e4i)T + (-1.08e8 + 1.00e8i)T^{2} \) |
| 29 | \( 1 + (-1.20e4 + 899. i)T + (5.88e8 - 8.86e7i)T^{2} \) |
| 31 | \( 1 + (-2.60e4 + 1.77e4i)T + (3.24e8 - 8.26e8i)T^{2} \) |
| 37 | \( 1 + (1.49e4 + 8.64e3i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-8.21e4 - 3.95e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-8.44e4 - 1.05e5i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-8.56e3 - 7.94e3i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-2.16e4 + 9.48e4i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.43e5 + 2.10e5i)T + (-1.88e10 - 4.79e10i)T^{2} \) |
| 67 | \( 1 + (3.87e5 - 5.83e4i)T + (8.64e10 - 2.66e10i)T^{2} \) |
| 71 | \( 1 + (-2.85e5 - 1.11e5i)T + (9.39e10 + 8.71e10i)T^{2} \) |
| 73 | \( 1 + (6.78e4 + 7.31e4i)T + (-1.13e10 + 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-2.02e5 - 3.50e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (6.21e4 - 8.28e5i)T + (-3.23e11 - 4.87e10i)T^{2} \) |
| 89 | \( 1 + (-3.36e5 - 2.51e4i)T + (4.91e11 + 7.40e10i)T^{2} \) |
| 97 | \( 1 + (-9.27e4 + 1.16e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−13.95614775562043338393984648468, −12.54879017660212143871114179281, −11.96933097196566312719432881727, −10.67532731012137380943035600657, −9.620550206012673488609455502923, −7.970646664968310426405919968910, −5.29640831384280601927354769241, −4.43766815776342483722397558629, −2.42042848476533082612876263074, −0.913882427743892198406947257264,
3.02224191671716552746474345255, 5.29427000647328844583395330952, 5.91313317960731655500446711769, 7.61609153947277080777496559891, 8.420976948785540714500386107575, 10.56571955656841379099661899044, 11.91883393412931732411472607138, 13.62098314450535295511790724427, 14.36649639826162829504371388396, 15.04304367923321138038456288072
