| L(s) = 1 | + (0.399 + 1.74i)2-s + (−2.95 − 2.74i)3-s + (4.30 − 2.07i)4-s + (12.0 − 1.81i)5-s + (3.62 − 6.27i)6-s + (7.86 + 13.6i)7-s + (14.3 + 17.9i)8-s + (−0.801 − 10.6i)9-s + (7.98 + 20.3i)10-s + (−19.4 − 9.38i)11-s + (−18.4 − 5.68i)12-s + (1.46 − 3.73i)13-s + (−20.6 + 19.2i)14-s + (−40.5 − 27.6i)15-s + (−1.83 + 2.29i)16-s + (−24.7 − 3.72i)17-s + ⋯ |
| L(s) = 1 | + (0.141 + 0.618i)2-s + (−0.569 − 0.528i)3-s + (0.538 − 0.259i)4-s + (1.07 − 0.162i)5-s + (0.246 − 0.426i)6-s + (0.424 + 0.735i)7-s + (0.631 + 0.792i)8-s + (−0.0296 − 0.395i)9-s + (0.252 + 0.643i)10-s + (−0.533 − 0.257i)11-s + (−0.443 − 0.136i)12-s + (0.0312 − 0.0797i)13-s + (−0.395 + 0.366i)14-s + (−0.698 − 0.476i)15-s + (−0.0286 + 0.0358i)16-s + (−0.352 − 0.0531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.56166 + 0.147545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56166 + 0.147545i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-85.2 - 268. i)T \) |
| good | 2 | \( 1 + (-0.399 - 1.74i)T + (-7.20 + 3.47i)T^{2} \) |
| 3 | \( 1 + (2.95 + 2.74i)T + (2.01 + 26.9i)T^{2} \) |
| 5 | \( 1 + (-12.0 + 1.81i)T + (119. - 36.8i)T^{2} \) |
| 7 | \( 1 + (-7.86 - 13.6i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (19.4 + 9.38i)T + (829. + 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-1.46 + 3.73i)T + (-1.61e3 - 1.49e3i)T^{2} \) |
| 17 | \( 1 + (24.7 + 3.72i)T + (4.69e3 + 1.44e3i)T^{2} \) |
| 19 | \( 1 + (-0.505 + 6.74i)T + (-6.78e3 - 1.02e3i)T^{2} \) |
| 23 | \( 1 + (147. - 100. i)T + (4.44e3 - 1.13e4i)T^{2} \) |
| 29 | \( 1 + (124. - 115. i)T + (1.82e3 - 2.43e4i)T^{2} \) |
| 31 | \( 1 + (40.5 + 12.5i)T + (2.46e4 + 1.67e4i)T^{2} \) |
| 37 | \( 1 + (-220. + 382. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (11.8 + 51.8i)T + (-6.20e4 + 2.99e4i)T^{2} \) |
| 47 | \( 1 + (-213. + 102. i)T + (6.47e4 - 8.11e4i)T^{2} \) |
| 53 | \( 1 + (34.1 + 87.0i)T + (-1.09e5 + 1.01e5i)T^{2} \) |
| 59 | \( 1 + (495. - 621. i)T + (-4.57e4 - 2.00e5i)T^{2} \) |
| 61 | \( 1 + (10.2 - 3.17i)T + (1.87e5 - 1.27e5i)T^{2} \) |
| 67 | \( 1 + (-62.0 + 828. i)T + (-2.97e5 - 4.48e4i)T^{2} \) |
| 71 | \( 1 + (384. + 262. i)T + (1.30e5 + 3.33e5i)T^{2} \) |
| 73 | \( 1 + (108. - 277. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (327. + 567. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-1.04e3 - 973. i)T + (4.27e4 + 5.70e5i)T^{2} \) |
| 89 | \( 1 + (238. + 221. i)T + (5.26e4 + 7.02e5i)T^{2} \) |
| 97 | \( 1 + (-1.61e3 - 777. i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−15.51556288687262047929198231711, −14.44658316197487477938030810843, −13.26927041831897327862144529886, −11.97222147301154066369161618854, −10.85634314270231999460419471927, −9.280439642755195226573950047232, −7.57188415482866790952307275700, −6.05694022007047609775102808742, −5.52651706855664715356455500601, −1.95082450850759448878758319196,
2.15535614339917663362478703266, 4.40981993227917362745125745825, 6.10537453296805783436126140293, 7.73958403953153355252080638644, 9.993455506494369896517511184011, 10.56824012858167149052690265577, 11.60821047267611637285639393885, 13.07404368409434266952449597473, 14.00671409985481655404179779656, 15.62968612227239345272998517613
