| L(s) = 1 | + (−3.52 + 7.31i)2-s + (12.1 − 17.8i)3-s + (−1.20 − 1.50i)4-s + (−66.5 − 71.7i)5-s + (87.7 + 151. i)6-s + (−41.7 − 24.0i)7-s + (−491. + 112. i)8-s + (95.6 + 243. i)9-s + (759. − 234. i)10-s + (−1.42e3 + 1.78e3i)11-s + (−41.5 + 3.11i)12-s + (−1.28e3 − 395. i)13-s + (323. − 220. i)14-s + (−2.09e3 + 315. i)15-s + (938. − 4.11e3i)16-s + (−768. − 713. i)17-s + ⋯ |
| L(s) = 1 | + (−0.440 + 0.914i)2-s + (0.450 − 0.661i)3-s + (−0.0187 − 0.0235i)4-s + (−0.532 − 0.574i)5-s + (0.406 + 0.703i)6-s + (−0.121 − 0.0702i)7-s + (−0.959 + 0.219i)8-s + (0.131 + 0.334i)9-s + (0.759 − 0.234i)10-s + (−1.06 + 1.33i)11-s + (−0.0240 + 0.00180i)12-s + (−0.583 − 0.179i)13-s + (0.117 − 0.0802i)14-s + (−0.619 + 0.0934i)15-s + (0.229 − 1.00i)16-s + (−0.156 − 0.145i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0427689 - 0.350112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0427689 - 0.350112i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.92e4 + 6.57e3i)T \) |
| good | 2 | \( 1 + (3.52 - 7.31i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-12.1 + 17.8i)T + (-266. - 678. i)T^{2} \) |
| 5 | \( 1 + (66.5 + 71.7i)T + (-1.16e3 + 1.55e4i)T^{2} \) |
| 7 | \( 1 + (41.7 + 24.0i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.42e3 - 1.78e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (1.28e3 + 395. i)T + (3.98e6 + 2.71e6i)T^{2} \) |
| 17 | \( 1 + (768. + 713. i)T + (1.80e6 + 2.40e7i)T^{2} \) |
| 19 | \( 1 + (5.89e3 + 2.31e3i)T + (3.44e7 + 3.19e7i)T^{2} \) |
| 23 | \( 1 + (1.60e4 + 2.42e3i)T + (1.41e8 + 4.36e7i)T^{2} \) |
| 29 | \( 1 + (-1.23e4 - 1.81e4i)T + (-2.17e8 + 5.53e8i)T^{2} \) |
| 31 | \( 1 + (-2.41e3 - 3.22e4i)T + (-8.77e8 + 1.32e8i)T^{2} \) |
| 37 | \( 1 + (4.07e4 - 2.35e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-5.65e4 - 2.72e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (5.77e4 + 7.23e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (6.41e4 - 1.97e4i)T + (1.83e10 - 1.24e10i)T^{2} \) |
| 59 | \( 1 + (4.75e4 - 2.08e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.39e5 - 1.04e4i)T + (5.09e10 + 7.67e9i)T^{2} \) |
| 67 | \( 1 + (-1.69e5 + 4.33e5i)T + (-6.63e10 - 6.15e10i)T^{2} \) |
| 71 | \( 1 + (-7.84e4 - 5.20e5i)T + (-1.22e11 + 3.77e10i)T^{2} \) |
| 73 | \( 1 + (1.04e5 - 3.39e5i)T + (-1.25e11 - 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-2.02e5 + 3.51e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (3.23e5 + 2.20e5i)T + (1.19e11 + 3.04e11i)T^{2} \) |
| 89 | \( 1 + (-3.41e5 + 5.01e5i)T + (-1.81e11 - 4.62e11i)T^{2} \) |
| 97 | \( 1 + (5.37e3 - 6.73e3i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−15.59969438757296542930698701471, −14.39465257730524253255582217904, −12.82883029610137551868625033642, −12.22408018111048754650893784374, −10.22119345708508681184044299886, −8.555046968621232482417125238730, −7.76295206235805321772130626127, −6.85466264393363306735783132575, −4.84706513918885804213300360380, −2.38055267813270576922567673735,
0.16629609928298672012841117011, 2.58696353372263124790438490163, 3.78875473585340107566038783473, 6.11817064474345548987835723337, 8.083110985417069523793959908082, 9.429957599794914933116672939957, 10.45390885572958570374953155522, 11.30143770643702157829693038551, 12.56913947576221509776948816591, 14.23719985813471536765590903946
