| L(s) = 1 | + (4.43 + 19.4i)2-s + (35.4 + 32.9i)3-s + (−241. + 116. i)4-s + (−172. + 26.0i)5-s + (−482. + 834. i)6-s + (17.2 + 29.8i)7-s + (−1.74e3 − 2.18e3i)8-s + (11.6 + 156. i)9-s + (−1.27e3 − 3.24e3i)10-s + (−1.20e3 − 580. i)11-s + (−1.24e4 − 3.83e3i)12-s + (−2.43e3 + 6.21e3i)13-s + (−503. + 466. i)14-s + (−6.99e3 − 4.76e3i)15-s + (1.33e4 − 1.66e4i)16-s + (7.69e3 + 1.16e3i)17-s + ⋯ |
| L(s) = 1 | + (0.391 + 1.71i)2-s + (0.758 + 0.704i)3-s + (−1.89 + 0.910i)4-s + (−0.618 + 0.0932i)5-s + (−0.911 + 1.57i)6-s + (0.0189 + 0.0328i)7-s + (−1.20 − 1.51i)8-s + (0.00534 + 0.0713i)9-s + (−0.402 − 1.02i)10-s + (−0.273 − 0.131i)11-s + (−2.07 − 0.640i)12-s + (−0.307 + 0.784i)13-s + (−0.0489 + 0.0454i)14-s + (−0.535 − 0.364i)15-s + (0.812 − 1.01i)16-s + (0.379 + 0.0572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.943465 - 1.34839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.943465 - 1.34839i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.07e5 + 1.19e5i)T \) |
| good | 2 | \( 1 + (-4.43 - 19.4i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (-35.4 - 32.9i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (172. - 26.0i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (-17.2 - 29.8i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.20e3 + 580. i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (2.43e3 - 6.21e3i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (-7.69e3 - 1.16e3i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (2.92e3 - 3.89e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (-2.03e4 + 1.38e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (-2.27e4 + 2.10e4i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (-1.14e5 - 3.52e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (1.31e5 - 2.27e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.56e5 - 6.86e5i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (5.33e5 - 2.56e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (-2.05e5 - 5.22e5i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (1.11e6 - 1.39e6i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (-8.84e5 + 2.72e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (1.10e5 - 1.47e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (1.24e6 + 8.51e5i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (-9.79e5 + 2.49e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (1.34e6 + 2.33e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-6.87e6 - 6.37e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (-5.31e6 - 4.93e6i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (-9.91e5 - 4.77e5i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−15.14811028251771071909063483764, −14.57756273032104301758698863553, −13.57788927162130931344657770965, −12.01293773157416721598778640448, −9.901755025672589829610115007964, −8.626368468095556687125465893294, −7.75216987267590277588748405004, −6.32946168970645906079540814304, −4.68674145044938907554867741657, −3.54992756640535637819084309424,
0.55914183679048151152482904432, 2.18728340859620587698278224204, 3.31782271340721523003311641498, 4.92979105669222452728416627482, 7.53528902407956952449546018401, 8.829992116995930105571319120230, 10.25631006893388887656421306012, 11.41248730794845134605939317824, 12.52068283407029295036217541719, 13.27360439697368483091949393638
