| L(s) = 1 | + (9.78 − 20.3i)2-s + (−69.5 + 101. i)3-s + (−157. − 197. i)4-s + (400. + 431. i)5-s + (1.39e3 + 2.41e3i)6-s + (−2.34e3 − 1.35e3i)7-s + (81.3 − 18.5i)8-s + (−3.17e3 − 8.07e3i)9-s + (1.26e4 − 3.91e3i)10-s + (2.31e3 − 2.89e3i)11-s + (3.10e4 − 2.32e3i)12-s + (−3.33e4 − 1.02e4i)13-s + (−5.04e4 + 3.43e4i)14-s + (−7.18e4 + 1.08e4i)15-s + (1.47e4 − 6.48e4i)16-s + (−5.71e4 − 5.30e4i)17-s + ⋯ |
| L(s) = 1 | + (0.611 − 1.26i)2-s + (−0.858 + 1.25i)3-s + (−0.614 − 0.770i)4-s + (0.640 + 0.690i)5-s + (1.07 + 1.85i)6-s + (−0.976 − 0.563i)7-s + (0.0198 − 0.00453i)8-s + (−0.483 − 1.23i)9-s + (1.26 − 0.391i)10-s + (0.157 − 0.197i)11-s + (1.49 − 0.112i)12-s + (−1.16 − 0.360i)13-s + (−1.31 + 0.894i)14-s + (−1.41 + 0.213i)15-s + (0.225 − 0.988i)16-s + (−0.684 − 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0477i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.998 + 0.0477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0126179 - 0.528646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0126179 - 0.528646i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.39e6 + 3.84e5i)T \) |
| good | 2 | \( 1 + (-9.78 + 20.3i)T + (-159. - 200. i)T^{2} \) |
| 3 | \( 1 + (69.5 - 101. i)T + (-2.39e3 - 6.10e3i)T^{2} \) |
| 5 | \( 1 + (-400. - 431. i)T + (-2.91e4 + 3.89e5i)T^{2} \) |
| 7 | \( 1 + (2.34e3 + 1.35e3i)T + (2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-2.31e3 + 2.89e3i)T + (-4.76e7 - 2.08e8i)T^{2} \) |
| 13 | \( 1 + (3.33e4 + 1.02e4i)T + (6.73e8 + 4.59e8i)T^{2} \) |
| 17 | \( 1 + (5.71e4 + 5.30e4i)T + (5.21e8 + 6.95e9i)T^{2} \) |
| 19 | \( 1 + (2.23e4 + 8.76e3i)T + (1.24e10 + 1.15e10i)T^{2} \) |
| 23 | \( 1 + (2.58e5 + 3.88e4i)T + (7.48e10 + 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-3.42e5 - 5.02e5i)T + (-1.82e11 + 4.65e11i)T^{2} \) |
| 31 | \( 1 + (-2.24e4 - 3.00e5i)T + (-8.43e11 + 1.27e11i)T^{2} \) |
| 37 | \( 1 + (-2.63e6 + 1.51e6i)T + (1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + (3.29e6 + 1.58e6i)T + (4.97e12 + 6.24e12i)T^{2} \) |
| 47 | \( 1 + (2.81e6 + 3.52e6i)T + (-5.29e12 + 2.32e13i)T^{2} \) |
| 53 | \( 1 + (8.24e6 - 2.54e6i)T + (5.14e13 - 3.50e13i)T^{2} \) |
| 59 | \( 1 + (2.62e6 - 1.14e7i)T + (-1.32e14 - 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-2.39e7 - 1.79e6i)T + (1.89e14 + 2.85e13i)T^{2} \) |
| 67 | \( 1 + (-2.16e6 + 5.51e6i)T + (-2.97e14 - 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-1.82e6 - 1.20e7i)T + (-6.17e14 + 1.90e14i)T^{2} \) |
| 73 | \( 1 + (4.54e6 - 1.47e7i)T + (-6.66e14 - 4.54e14i)T^{2} \) |
| 79 | \( 1 + (1.87e7 - 3.24e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-7.21e7 - 4.91e7i)T + (8.22e14 + 2.09e15i)T^{2} \) |
| 89 | \( 1 + (5.17e7 - 7.58e7i)T + (-1.43e15 - 3.66e15i)T^{2} \) |
| 97 | \( 1 + (-3.51e7 + 4.40e7i)T + (-1.74e15 - 7.64e15i)T^{2} \) |
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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
−13.44513499646440350039424786178, −12.18988860250507020975802522694, −11.05940707708513718667319765529, −10.16027482838589585639772693350, −9.749573799582942775380750354412, −6.71105400679640325754387097962, −5.14519338868550011599267849747, −3.89978320811406938369875551379, −2.62505966185380561333952394020, −0.17011502345292974189290158210,
1.84373851964072689500612574705, 4.86120731949229125586707558593, 6.10419572281437878076200231999, 6.59917355789080682535013675064, 8.001002932994585717461438155180, 9.697678635128127294775771807184, 11.81356234984489641650582060573, 12.89321573366282873770764636397, 13.33036121505557048896396306580, 14.77194930898600523491090016468
