| L(s) = 1 | + (−13.3 + 16.7i)2-s + (9.91 − 1.49i)3-s + (−74.0 − 324. i)4-s + (247. − 168. i)5-s + (−107. + 186. i)6-s + (−560. − 970. i)7-s + (3.96e3 + 1.90e3i)8-s + (−1.99e3 + 615. i)9-s + (−480. + 6.40e3i)10-s + (−736. + 3.22e3i)11-s + (−1.21e3 − 3.10e3i)12-s + (725. + 9.67e3i)13-s + (2.37e4 + 3.58e3i)14-s + (2.19e3 − 2.04e3i)15-s + (−4.67e4 + 2.24e4i)16-s + (2.01e3 + 1.37e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 + 1.48i)2-s + (0.211 − 0.0319i)3-s + (−0.578 − 2.53i)4-s + (0.884 − 0.603i)5-s + (−0.203 + 0.352i)6-s + (−0.617 − 1.06i)7-s + (2.73 + 1.31i)8-s + (−0.911 + 0.281i)9-s + (−0.151 + 2.02i)10-s + (−0.166 + 0.730i)11-s + (−0.203 − 0.519i)12-s + (0.0915 + 1.22i)13-s + (2.31 + 0.349i)14-s + (0.168 − 0.156i)15-s + (−2.85 + 1.37i)16-s + (0.0995 + 0.0678i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0552i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0145222 + 0.525374i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0145222 + 0.525374i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.46e4 - 5.17e5i)T \) |
| good | 2 | \( 1 + (13.3 - 16.7i)T + (-28.4 - 124. i)T^{2} \) |
| 3 | \( 1 + (-9.91 + 1.49i)T + (2.08e3 - 644. i)T^{2} \) |
| 5 | \( 1 + (-247. + 168. i)T + (2.85e4 - 7.27e4i)T^{2} \) |
| 7 | \( 1 + (560. + 970. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (736. - 3.22e3i)T + (-1.75e7 - 8.45e6i)T^{2} \) |
| 13 | \( 1 + (-725. - 9.67e3i)T + (-6.20e7 + 9.35e6i)T^{2} \) |
| 17 | \( 1 + (-2.01e3 - 1.37e3i)T + (1.49e8 + 3.81e8i)T^{2} \) |
| 19 | \( 1 + (2.89e4 + 8.94e3i)T + (7.38e8 + 5.03e8i)T^{2} \) |
| 23 | \( 1 + (-8.36e4 - 7.75e4i)T + (2.54e8 + 3.39e9i)T^{2} \) |
| 29 | \( 1 + (6.57e4 + 9.91e3i)T + (1.64e10 + 5.08e9i)T^{2} \) |
| 31 | \( 1 + (-9.62e4 - 2.45e5i)T + (-2.01e10 + 1.87e10i)T^{2} \) |
| 37 | \( 1 + (-8.43e4 + 1.46e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (6.89e4 - 8.64e4i)T + (-4.33e10 - 1.89e11i)T^{2} \) |
| 47 | \( 1 + (1.02e5 + 4.47e5i)T + (-4.56e11 + 2.19e11i)T^{2} \) |
| 53 | \( 1 + (6.57e4 - 8.77e5i)T + (-1.16e12 - 1.75e11i)T^{2} \) |
| 59 | \( 1 + (2.67e6 - 1.28e6i)T + (1.55e12 - 1.94e12i)T^{2} \) |
| 61 | \( 1 + (-5.78e5 + 1.47e6i)T + (-2.30e12 - 2.13e12i)T^{2} \) |
| 67 | \( 1 + (3.26e6 + 1.00e6i)T + (5.00e12 + 3.41e12i)T^{2} \) |
| 71 | \( 1 + (-4.01e5 + 3.72e5i)T + (6.79e11 - 9.06e12i)T^{2} \) |
| 73 | \( 1 + (1.74e5 + 2.33e6i)T + (-1.09e13 + 1.64e12i)T^{2} \) |
| 79 | \( 1 + (-3.18e6 - 5.51e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-5.59e5 + 8.42e4i)T + (2.59e13 - 7.99e12i)T^{2} \) |
| 89 | \( 1 + (2.04e6 - 3.07e5i)T + (4.22e13 - 1.30e13i)T^{2} \) |
| 97 | \( 1 + (1.48e6 - 6.51e6i)T + (-7.27e13 - 3.50e13i)T^{2} \) |
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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
−15.24282567586030465961789185084, −14.07267673406411673137160027051, −13.32273939772584111755654178691, −10.79467774052968736839482536614, −9.558415208574754521234177336888, −8.908701713998860862975851026072, −7.41492663392357795559816907543, −6.37310661807600666940219622903, −4.95942776025692694500909708763, −1.45101863248771442600505176681,
0.32175739334881122266453203767, 2.46150241205238158733486584817, 3.07457124238940191650941385968, 6.05914469419501291434174788314, 8.286837621710960940655917284872, 9.106428182766676922531980434290, 10.22918333044226362944401395328, 11.13873193093341626077786860516, 12.43904416814909587611724360633, 13.36399039168190746686796380533
