| L(s) = 1 | + (−10.2 − 21.3i)2-s + (22.6 − 47.0i)3-s + (−189. + 237. i)4-s + (67.6 + 15.4i)5-s − 1.23e3·6-s + 643. i·7-s + (1.10e3 + 252. i)8-s + (2.38e3 + 2.99e3i)9-s + (−365. − 1.60e3i)10-s + (1.10e4 + 1.38e4i)11-s + (6.88e3 + 1.43e4i)12-s + (−6.52e3 + 2.85e4i)13-s + (1.37e4 − 6.60e3i)14-s + (2.26e3 − 2.83e3i)15-s + (1.13e4 + 4.96e4i)16-s + (1.70e4 + 7.46e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.641 − 1.33i)2-s + (0.279 − 0.581i)3-s + (−0.740 + 0.928i)4-s + (0.108 + 0.0247i)5-s − 0.953·6-s + 0.268i·7-s + (0.270 + 0.0616i)8-s + (0.364 + 0.456i)9-s + (−0.0365 − 0.160i)10-s + (0.754 + 0.945i)11-s + (0.332 + 0.689i)12-s + (−0.228 + 1.00i)13-s + (0.357 − 0.171i)14-s + (0.0446 − 0.0559i)15-s + (0.173 + 0.758i)16-s + (0.203 + 0.893i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.18401 - 0.281983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18401 - 0.281983i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.41e6 - 2.05e5i)T \) |
| good | 2 | \( 1 + (10.2 + 21.3i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (-22.6 + 47.0i)T + (-4.09e3 - 5.12e3i)T^{2} \) |
| 5 | \( 1 + (-67.6 - 15.4i)T + (3.51e5 + 1.69e5i)T^{2} \) |
| 7 | \( 1 - 643. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.10e4 - 1.38e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (6.52e3 - 2.85e4i)T + (-7.34e8 - 3.53e8i)T^{2} \) |
| 17 | \( 1 + (-1.70e4 - 7.46e4i)T + (-6.28e9 + 3.02e9i)T^{2} \) |
| 19 | \( 1 + (1.06e4 + 8.51e3i)T + (3.77e9 + 1.65e10i)T^{2} \) |
| 23 | \( 1 + (1.16e5 + 1.46e5i)T + (-1.74e10 + 7.63e10i)T^{2} \) |
| 29 | \( 1 + (-7.54e4 - 1.56e5i)T + (-3.11e11 + 3.91e11i)T^{2} \) |
| 31 | \( 1 + (-2.67e5 + 1.28e5i)T + (5.31e11 - 6.66e11i)T^{2} \) |
| 37 | \( 1 - 8.90e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (1.32e6 - 6.39e5i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (1.33e6 - 1.67e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-2.62e6 - 1.15e7i)T + (-5.60e13 + 2.70e13i)T^{2} \) |
| 59 | \( 1 + (-1.98e6 - 8.68e6i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-5.06e6 + 1.05e7i)T + (-1.19e14 - 1.49e14i)T^{2} \) |
| 67 | \( 1 + (1.35e6 - 1.69e6i)T + (-9.03e13 - 3.95e14i)T^{2} \) |
| 71 | \( 1 + (-7.25e6 - 5.78e6i)T + (1.43e14 + 6.29e14i)T^{2} \) |
| 73 | \( 1 + (-1.00e7 - 2.29e6i)T + (7.26e14 + 3.49e14i)T^{2} \) |
| 79 | \( 1 - 5.41e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-3.52e7 - 1.69e7i)T + (1.40e15 + 1.76e15i)T^{2} \) |
| 89 | \( 1 + (2.53e7 - 5.26e7i)T + (-2.45e15 - 3.07e15i)T^{2} \) |
| 97 | \( 1 + (8.26e6 + 1.03e7i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−13.77338461460251048717781516383, −12.50372276455477057802139284275, −11.85117999330202653244809953849, −10.41811351286601137196413722818, −9.453072943383398084252062869548, −8.200697621612891915400888428950, −6.61209704855015697234396281570, −4.18844722902282120353411161352, −2.28211209115665077286107415926, −1.46120733206397660120087552214,
0.58939722474451188888051407702, 3.50800819344224584141841387471, 5.41642978981688152736392694938, 6.73486610305362701966235233417, 8.019808683128322198323199801986, 9.178740349991778033323894414577, 10.08150578090997937428502983451, 11.83595858856610804840011174496, 13.63844118704206963874065413542, 14.71662758733325483712977963421
