| L(s) = 1 | + (−2.15 − 4.46i)2-s + (77.0 + 113. i)3-s + (144. − 180. i)4-s + (−373. + 402. i)5-s + (339. − 587. i)6-s + (−2.25e3 + 1.30e3i)7-s + (−2.35e3 − 537. i)8-s + (−4.44e3 + 1.13e4i)9-s + (2.59e3 + 801. i)10-s + (−3.62e3 − 4.55e3i)11-s + (3.15e4 + 2.36e3i)12-s + (−2.63e4 + 8.12e3i)13-s + (1.06e4 + 7.27e3i)14-s + (−7.42e4 − 1.11e4i)15-s + (−1.05e4 − 4.60e4i)16-s + (3.00e4 − 2.79e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.134 − 0.279i)2-s + (0.951 + 1.39i)3-s + (0.563 − 0.706i)4-s + (−0.596 + 0.643i)5-s + (0.261 − 0.453i)6-s + (−0.939 + 0.542i)7-s + (−0.575 − 0.131i)8-s + (−0.677 + 1.72i)9-s + (0.259 + 0.0801i)10-s + (−0.247 − 0.310i)11-s + (1.52 + 0.114i)12-s + (−0.922 + 0.284i)13-s + (0.277 + 0.189i)14-s + (−1.46 − 0.220i)15-s + (−0.160 − 0.702i)16-s + (0.360 − 0.334i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.193996 + 1.12708i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.193996 + 1.12708i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.36e6 + 6.14e5i)T \) |
| good | 2 | \( 1 + (2.15 + 4.46i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (-77.0 - 113. i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (373. - 402. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (2.25e3 - 1.30e3i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (3.62e3 + 4.55e3i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (2.63e4 - 8.12e3i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (-3.00e4 + 2.79e4i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (-9.37e4 + 3.68e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (2.97e5 - 4.48e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (3.77e5 - 5.54e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (1.04e5 - 1.38e6i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (1.01e6 + 5.83e5i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-4.44e6 + 2.13e6i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-9.86e4 + 1.23e5i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-5.23e6 - 1.61e6i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (-4.60e6 - 2.01e7i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (2.89e6 - 2.16e5i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (-5.70e6 - 1.45e7i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-3.12e6 + 2.07e7i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (-9.39e6 - 3.04e7i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (-3.36e7 - 5.83e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-6.10e7 + 4.16e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (4.68e7 + 6.87e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (-3.97e7 - 4.98e7i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−14.87555917328788574757077402043, −14.09754341712050945845693473666, −12.05774812527970707216033663016, −10.77416677407110383427030212988, −9.880155082438386042035132642369, −9.049005311630094094810290060908, −7.22259581649430246921063545842, −5.37985263589314649644961063572, −3.47682493305954594685237560352, −2.60426084049728278282591719313,
0.36567869617863895037719050373, 2.27996960694404809579563332515, 3.63194613784336967330361845693, 6.41406398541282708229832382215, 7.65933198401900312019641967786, 8.008776832988655472217109865751, 9.624965151107936756372978941271, 11.91627294608383710483337537655, 12.59376254411848687385570275568, 13.39185914734608516155431187040
