| L(s) = 1 | + (0.0251 + 0.110i)2-s + (−21.2 − 19.7i)3-s + (115. − 55.5i)4-s + (9.70 − 1.46i)5-s + (1.63 − 2.83i)6-s + (672. + 1.16e3i)7-s + (18.0 + 22.5i)8-s + (−100. − 1.34e3i)9-s + (0.404 + 1.03i)10-s + (−330. − 159. i)11-s + (−3.54e3 − 1.09e3i)12-s + (1.51e3 − 3.86e3i)13-s + (−111. + 103. i)14-s + (−235. − 160. i)15-s + (1.02e4 − 1.28e4i)16-s + (2.39e4 + 3.61e3i)17-s + ⋯ |
| L(s) = 1 | + (0.00222 + 0.00972i)2-s + (−0.454 − 0.421i)3-s + (0.900 − 0.433i)4-s + (0.0347 − 0.00523i)5-s + (0.00309 − 0.00535i)6-s + (0.741 + 1.28i)7-s + (0.0124 + 0.0156i)8-s + (−0.0460 − 0.614i)9-s + (0.000128 + 0.000326i)10-s + (−0.0748 − 0.0360i)11-s + (−0.592 − 0.182i)12-s + (0.191 − 0.487i)13-s + (−0.0108 + 0.0100i)14-s + (−0.0179 − 0.0122i)15-s + (0.623 − 0.781i)16-s + (1.18 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.753i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.656 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.90403 - 0.866446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90403 - 0.866446i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.49e5 - 3.86e5i)T \) |
| good | 2 | \( 1 + (-0.0251 - 0.110i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (21.2 + 19.7i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (-9.70 + 1.46i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (-672. - 1.16e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (330. + 159. i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (-1.51e3 + 3.86e3i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (-2.39e4 - 3.61e3i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (-1.95e3 + 2.60e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (-8.29e4 + 5.65e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (-1.29e5 + 1.20e5i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (2.03e5 + 6.28e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (-2.53e5 + 4.39e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.74e5 - 7.65e5i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (4.44e5 - 2.14e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (-5.07e5 - 1.29e6i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (5.98e5 - 7.50e5i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (-1.52e6 + 4.69e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (8.99e4 - 1.19e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (2.16e6 + 1.47e6i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (-1.15e6 + 2.93e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (-2.46e6 - 4.27e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-5.93e6 - 5.50e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (6.46e5 + 5.99e5i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (7.33e6 + 3.53e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−14.73596979477496575458494610375, −12.77928839201481555676530786821, −11.80395122739034590511592220959, −11.03172411034109005591164310871, −9.383107357520950756825195033910, −7.80591730472378241465500270383, −6.29635119600522826212501882165, −5.37873662865680957447084657811, −2.67945820230715562286633334388, −1.06947494180039753536906781168,
1.51328256437919454889229311271, 3.66200050665781718160182310513, 5.27379410560643897079875800363, 7.07663920887356404484373777598, 8.023232959389318341137089277092, 10.17503087012632335531733178113, 10.98924829102600520765798833727, 11.92156417787895026158770244793, 13.50091871899019100113626747335, 14.62736722955661984780831862985
