| L(s) = 1 | − 17.9·2-s + (33.1 − 57.4i)3-s + 192.·4-s + (49.7 − 86.2i)5-s + (−594. + 1.02e3i)6-s + (775. + 1.34e3i)7-s − 1.15e3·8-s + (−1.10e3 − 1.92e3i)9-s + (−891. + 1.54e3i)10-s − 7.04e3·11-s + (6.39e3 − 1.10e4i)12-s + (4.42e3 + 7.66e3i)13-s + (−1.38e4 − 2.40e4i)14-s + (−3.30e3 − 5.72e3i)15-s − 3.92e3·16-s + (1.75e4 + 3.04e4i)17-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + (0.709 − 1.22i)3-s + 1.50·4-s + (0.178 − 0.308i)5-s + (−1.12 + 1.94i)6-s + (0.854 + 1.48i)7-s − 0.799·8-s + (−0.507 − 0.878i)9-s + (−0.281 + 0.488i)10-s − 1.59·11-s + (1.06 − 1.84i)12-s + (0.558 + 0.968i)13-s + (−1.35 − 2.34i)14-s + (−0.252 − 0.437i)15-s − 0.239·16-s + (0.868 + 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.939352 + 0.243055i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.939352 + 0.243055i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.65e5 + 4.94e5i)T \) |
| good | 2 | \( 1 + 17.9T + 128T^{2} \) |
| 3 | \( 1 + (-33.1 + 57.4i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-49.7 + 86.2i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-775. - 1.34e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + 7.04e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.42e3 - 7.66e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.75e4 - 3.04e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.02e4 - 1.77e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.25e4 - 2.16e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (7.51e4 + 1.30e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (5.13e4 - 8.90e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-9.74e4 + 1.68e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 3.24e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 2.06e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (8.19e4 - 1.41e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + 6.26e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + (2.37e5 + 4.11e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.82e6 - 3.15e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.98e5 + 5.16e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.04e6 + 1.80e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.72e6 - 4.71e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.93e6 + 3.34e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-7.75e5 + 1.34e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.59e7T + 8.07e13T^{2} \) |
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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
−14.67416149627732719318433143274, −13.18691309245653192807942556567, −12.11607713080042581367020863030, −10.72572796952968019631892601922, −9.088487887197328989976020071727, −8.281325305825404538658369057450, −7.65046749456815706229861515814, −5.82251251285117937572998594635, −2.24089533689520581333914327425, −1.52500593353261324724224810313,
0.65164640996227796758486493192, 2.86500652586144633567828859968, 4.76593676182891181657631364373, 7.45717936313681808908759230746, 8.191961871169739346738483596933, 9.579292069698187494817989859014, 10.58469373470960964562995759402, 10.81484755527672533183429439492, 13.50443369147134897990986707591, 14.65577805286840207304030958161
