| L(s) = 1 | + (0.293 − 1.28i)2-s + (0.969 + 4.24i)3-s + (5.63 + 2.71i)4-s + (−9.73 + 12.2i)5-s + 5.75·6-s + 4.97·7-s + (11.7 − 14.7i)8-s + (7.23 − 3.48i)9-s + (12.8 + 16.1i)10-s + (−3.36 + 1.62i)11-s + (−6.06 + 26.5i)12-s + (21.6 − 27.1i)13-s + (1.46 − 6.40i)14-s + (−61.2 − 29.4i)15-s + (15.7 + 19.7i)16-s + (−56.2 − 70.5i)17-s + ⋯ |
| L(s) = 1 | + (0.103 − 0.455i)2-s + (0.186 + 0.817i)3-s + (0.704 + 0.339i)4-s + (−0.870 + 1.09i)5-s + 0.391·6-s + 0.268·7-s + (0.518 − 0.650i)8-s + (0.268 − 0.129i)9-s + (0.406 + 0.509i)10-s + (−0.0922 + 0.0444i)11-s + (−0.145 + 0.639i)12-s + (0.461 − 0.578i)13-s + (0.0279 − 0.122i)14-s + (−1.05 − 0.507i)15-s + (0.245 + 0.307i)16-s + (−0.802 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.46504 + 0.539124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46504 + 0.539124i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (18.4 + 281. i)T \) |
| good | 2 | \( 1 + (-0.293 + 1.28i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-0.969 - 4.24i)T + (-24.3 + 11.7i)T^{2} \) |
| 5 | \( 1 + (9.73 - 12.2i)T + (-27.8 - 121. i)T^{2} \) |
| 7 | \( 1 - 4.97T + 343T^{2} \) |
| 11 | \( 1 + (3.36 - 1.62i)T + (829. - 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-21.6 + 27.1i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (56.2 + 70.5i)T + (-1.09e3 + 4.78e3i)T^{2} \) |
| 19 | \( 1 + (17.3 + 8.35i)T + (4.27e3 + 5.36e3i)T^{2} \) |
| 23 | \( 1 + (30.8 - 14.8i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-6.51 + 28.5i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-49.9 + 218. i)T + (-2.68e4 - 1.29e4i)T^{2} \) |
| 37 | \( 1 - 214.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (108. - 475. i)T + (-6.20e4 - 2.99e4i)T^{2} \) |
| 47 | \( 1 + (-103. - 49.8i)T + (6.47e4 + 8.11e4i)T^{2} \) |
| 53 | \( 1 + (426. + 534. i)T + (-3.31e4 + 1.45e5i)T^{2} \) |
| 59 | \( 1 + (231. + 289. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-190. - 832. i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 + (-428. - 206. i)T + (1.87e5 + 2.35e5i)T^{2} \) |
| 71 | \( 1 + (792. + 381. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (418. - 524. i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 + 115.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-209. - 916. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-133. - 583. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + (598. - 288. i)T + (5.69e5 - 7.13e5i)T^{2} \) |
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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
−15.50562607117027749869684923025, −14.85775946618672158936663033686, −13.15430904790365997920620353619, −11.60400585018174493701906533339, −10.99359941940307072070356496301, −9.869882411143851346750044856992, −7.928844780590470238674868384144, −6.73637043239142353026220093509, −4.18316610648378607257289634912, −2.93808212653750608784101907146,
1.58858633645776896960441850418, 4.57351689104229346342932448609, 6.39940871540865664410995667850, 7.68720233377669484893631184363, 8.611300590966086204767365535261, 10.74774703947073999301766787245, 11.98384378969013576410787238919, 12.93914299399928066053733748248, 14.21586347156498947479187986992, 15.58701843571179369426349936586
