| L(s) = 1 | + (5.45 + 11.3i)2-s + (−9.73 − 14.2i)3-s + (−58.6 + 73.5i)4-s + (−58.7 + 63.3i)5-s + (108. − 188. i)6-s + (−290. + 167. i)7-s + (−368. − 84.1i)8-s + (157. − 400. i)9-s + (−1.03e3 − 320. i)10-s + (−1.32e3 − 1.66e3i)11-s + (1.62e3 + 121. i)12-s + (−1.68e3 + 518. i)13-s + (−3.48e3 − 2.37e3i)14-s + (1.47e3 + 222. i)15-s + (281. + 1.23e3i)16-s + (−1.04e3 + 971. i)17-s + ⋯ |
| L(s) = 1 | + (0.681 + 1.41i)2-s + (−0.360 − 0.528i)3-s + (−0.916 + 1.14i)4-s + (−0.470 + 0.506i)5-s + (0.503 − 0.871i)6-s + (−0.848 + 0.489i)7-s + (−0.720 − 0.164i)8-s + (0.215 − 0.549i)9-s + (−1.03 − 0.320i)10-s + (−0.997 − 1.25i)11-s + (0.938 + 0.0703i)12-s + (−0.764 + 0.235i)13-s + (−1.27 − 0.866i)14-s + (0.437 + 0.0659i)15-s + (0.0687 + 0.301i)16-s + (−0.213 + 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.229344 - 0.401141i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.229344 - 0.401141i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.88e4 + 7.40e4i)T \) |
| good | 2 | \( 1 + (-5.45 - 11.3i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (9.73 + 14.2i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (58.7 - 63.3i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (290. - 167. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.32e3 + 1.66e3i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (1.68e3 - 518. i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (1.04e3 - 971. i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (4.16e3 - 1.63e3i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (-1.37e4 + 2.07e3i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (1.32e4 - 1.94e4i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (2.78e3 - 3.71e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (2.26e4 + 1.30e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-3.40e3 + 1.64e3i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (2.68e4 - 3.36e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (4.09e3 + 1.26e3i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (-4.85e3 - 2.12e4i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-3.04e5 + 2.28e4i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (1.82e5 + 4.65e5i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (4.47e4 - 2.96e5i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-1.07e4 - 3.49e4i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (2.64e5 + 4.58e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.81e5 - 1.23e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (-5.31e5 - 7.80e5i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-7.73e4 - 9.69e4i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−15.45925932349871948248799024125, −14.56758034743332462574667441423, −13.20127041203747970614375262065, −12.46595424968501862005624032279, −10.84699336939066868458795228091, −8.827374760835088963191624217874, −7.34966393426004592561638541613, −6.50805601769022161550522438708, −5.37510186665489628958071259739, −3.36708545942961567256757087039,
0.17321890591300699542442949877, 2.42934619328992613634668926113, 4.20951217278747614292593882344, 5.02668120254614888599625769311, 7.47913115436281143000253954358, 9.726959096870658063018447750765, 10.37353375476394605952862754464, 11.54295119693908036230810794392, 12.84979275646945951529543703699, 13.17948467711505661257916609785
