| L(s) = 1 | + (2.46 + 10.7i)2-s + (15.0 + 13.9i)3-s + (−81.6 + 39.3i)4-s + (69.8 − 10.5i)5-s + (−113. + 197. i)6-s + (10.5 + 18.2i)7-s + (−404. − 507. i)8-s + (13.4 + 179. i)9-s + (285. + 727. i)10-s + (342. + 164. i)11-s + (−1.78e3 − 549. i)12-s + (402. − 1.02e3i)13-s + (−171. + 158. i)14-s + (1.19e3 + 817. i)15-s + (2.67e3 − 3.35e3i)16-s + (−761. − 114. i)17-s + ⋯ |
| L(s) = 1 | + (0.435 + 1.90i)2-s + (0.966 + 0.897i)3-s + (−2.55 + 1.22i)4-s + (1.24 − 0.188i)5-s + (−1.29 + 2.23i)6-s + (0.0814 + 0.140i)7-s + (−2.23 − 2.80i)8-s + (0.0552 + 0.737i)9-s + (0.903 + 2.30i)10-s + (0.853 + 0.410i)11-s + (−3.56 − 1.10i)12-s + (0.661 − 1.68i)13-s + (−0.233 + 0.216i)14-s + (1.37 + 0.938i)15-s + (2.61 − 3.27i)16-s + (−0.639 − 0.0963i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0854585 - 2.69705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0854585 - 2.69705i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.19e3 - 1.19e4i)T \) |
| good | 2 | \( 1 + (-2.46 - 10.7i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (-15.0 - 13.9i)T + (18.1 + 242. i)T^{2} \) |
| 5 | \( 1 + (-69.8 + 10.5i)T + (2.98e3 - 921. i)T^{2} \) |
| 7 | \( 1 + (-10.5 - 18.2i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-342. - 164. i)T + (1.00e5 + 1.25e5i)T^{2} \) |
| 13 | \( 1 + (-402. + 1.02e3i)T + (-2.72e5 - 2.52e5i)T^{2} \) |
| 17 | \( 1 + (761. + 114. i)T + (1.35e6 + 4.18e5i)T^{2} \) |
| 19 | \( 1 + (-13.9 + 185. i)T + (-2.44e6 - 3.69e5i)T^{2} \) |
| 23 | \( 1 + (2.32e3 - 1.58e3i)T + (2.35e6 - 5.99e6i)T^{2} \) |
| 29 | \( 1 + (3.53e3 - 3.28e3i)T + (1.53e6 - 2.04e7i)T^{2} \) |
| 31 | \( 1 + (313. + 96.7i)T + (2.36e7 + 1.61e7i)T^{2} \) |
| 37 | \( 1 + (749. - 1.29e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-533. - 2.33e3i)T + (-1.04e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (8.34e3 - 4.02e3i)T + (1.42e8 - 1.79e8i)T^{2} \) |
| 53 | \( 1 + (1.33e4 + 3.41e4i)T + (-3.06e8 + 2.84e8i)T^{2} \) |
| 59 | \( 1 + (-481. + 603. i)T + (-1.59e8 - 6.96e8i)T^{2} \) |
| 61 | \( 1 + (5.94e3 - 1.83e3i)T + (6.97e8 - 4.75e8i)T^{2} \) |
| 67 | \( 1 + (-2.20e3 + 2.94e4i)T + (-1.33e9 - 2.01e8i)T^{2} \) |
| 71 | \( 1 + (-3.94e4 - 2.68e4i)T + (6.59e8 + 1.67e9i)T^{2} \) |
| 73 | \( 1 + (-8.57e3 + 2.18e4i)T + (-1.51e9 - 1.41e9i)T^{2} \) |
| 79 | \( 1 + (1.25e4 + 2.16e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-4.30e4 - 3.99e4i)T + (2.94e8 + 3.92e9i)T^{2} \) |
| 89 | \( 1 + (-1.16e4 - 1.07e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + (-7.72e4 - 3.72e4i)T + (5.35e9 + 6.71e9i)T^{2} \) |
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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
−15.34840550705601059707146945331, −14.64467976317628970838849284369, −13.73251434342103669549393949648, −12.89730769833665530915230398593, −9.855154915345331596950782692406, −9.105244483716405560646815625419, −8.063517926508498512978868971956, −6.31500880306729593812044514741, −5.15469665561617718886783204576, −3.61923406286716003314176474856,
1.53799286176915786125957999252, 2.28589510768424831057558570579, 4.01542787637047116849288657501, 6.20163354571457849347705418751, 8.760990961220606170813258777424, 9.469971029728069815160226075135, 10.89163810439096449307446324604, 12.08123526967424043943554370571, 13.33983854302700240235641658962, 13.91323027412592803702408619692
