| L(s) = 1 | + (0.645 + 1.34i)2-s + (42.1 + 61.8i)3-s + (158. − 198. i)4-s + (−171. + 184. i)5-s + (−55.6 + 96.4i)6-s + (2.10e3 − 1.21e3i)7-s + (739. + 168. i)8-s + (350. − 891. i)9-s + (−357. − 110. i)10-s + (−7.36e3 − 9.23e3i)11-s + (1.89e4 + 1.41e3i)12-s + (2.84e4 − 8.77e3i)13-s + (2.98e3 + 2.03e3i)14-s + (−1.86e4 − 2.80e3i)15-s + (−1.42e4 − 6.22e4i)16-s + (−1.14e5 + 1.05e5i)17-s + ⋯ |
| L(s) = 1 | + (0.0403 + 0.0837i)2-s + (0.520 + 0.763i)3-s + (0.618 − 0.775i)4-s + (−0.273 + 0.295i)5-s + (−0.0429 + 0.0744i)6-s + (0.874 − 0.505i)7-s + (0.180 + 0.0412i)8-s + (0.0533 − 0.135i)9-s + (−0.0357 − 0.0110i)10-s + (−0.502 − 0.630i)11-s + (0.913 + 0.0684i)12-s + (0.995 − 0.307i)13-s + (0.0776 + 0.0529i)14-s + (−0.367 − 0.0554i)15-s + (−0.216 − 0.949i)16-s + (−1.36 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.80578 - 0.220550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.80578 - 0.220550i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.41e6 + 1.16e4i)T \) |
| good | 2 | \( 1 + (-0.645 - 1.34i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (-42.1 - 61.8i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (171. - 184. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-2.10e3 + 1.21e3i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (7.36e3 + 9.23e3i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-2.84e4 + 8.77e3i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (1.14e5 - 1.05e5i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (-1.79e5 + 7.03e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (-3.32e5 + 5.01e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-2.29e5 + 3.36e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (-2.32e4 + 3.09e5i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (-1.03e6 - 5.98e5i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (1.22e6 - 5.91e5i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (5.47e6 - 6.87e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (1.38e7 + 4.27e6i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (2.03e6 + 8.91e6i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (1.87e7 - 1.40e6i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (1.09e6 + 2.78e6i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-6.81e6 + 4.51e7i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (-1.33e7 - 4.34e7i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (7.94e6 + 1.37e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (9.53e6 - 6.49e6i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (1.38e7 + 2.02e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (-9.98e7 - 1.25e8i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−14.48131897874547333945857977201, −13.33809223318048311398961151695, −11.10537191327118485895577516291, −10.88261521328692355444472341884, −9.336428658420790900920972797121, −7.936599281453029367509920937303, −6.37746213511590946608527145730, −4.73659381394076777521920337624, −3.16040280582910824219119286122, −1.16708956722798179031389611005,
1.58063103327394437434713825413, 2.79824840465911984875808685096, 4.78781511629505870653152825995, 6.95047630117469797480126861234, 7.88436520580501021197518686291, 8.852151186343395676511127441500, 11.02403988065937448978071313508, 11.94867133693540861009695892212, 13.01958589249673137649784220980, 13.98685422135374097746315355405
