| L(s) = 1 | + (−1.52 − 3.17i)2-s + (2.15 − 4.48i)3-s + (−5.22 + 6.55i)4-s + (5.44 + 1.24i)5-s − 17.5·6-s + 9.10i·7-s + (15.0 + 3.43i)8-s + (−9.83 − 12.3i)9-s + (−4.37 − 19.1i)10-s + (−0.0631 − 0.0791i)11-s + (18.1 + 37.6i)12-s + (0.299 − 1.31i)13-s + (28.8 − 13.8i)14-s + (17.3 − 21.7i)15-s + (−4.62 − 20.2i)16-s + (3.77 + 16.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.763 − 1.58i)2-s + (0.719 − 1.49i)3-s + (−1.30 + 1.63i)4-s + (1.08 + 0.248i)5-s − 2.92·6-s + 1.30i·7-s + (1.88 + 0.429i)8-s + (−1.09 − 1.37i)9-s + (−0.437 − 1.91i)10-s + (−0.00573 − 0.00719i)11-s + (1.50 + 3.13i)12-s + (0.0230 − 0.100i)13-s + (2.06 − 0.992i)14-s + (1.15 − 1.44i)15-s + (−0.289 − 1.26i)16-s + (0.221 + 0.971i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.536i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.271389 - 0.933275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.271389 - 0.933275i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (19.7 - 38.2i)T \) |
| good | 2 | \( 1 + (1.52 + 3.17i)T + (-2.49 + 3.12i)T^{2} \) |
| 3 | \( 1 + (-2.15 + 4.48i)T + (-5.61 - 7.03i)T^{2} \) |
| 5 | \( 1 + (-5.44 - 1.24i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 - 9.10iT - 49T^{2} \) |
| 11 | \( 1 + (0.0631 + 0.0791i)T + (-26.9 + 117. i)T^{2} \) |
| 13 | \( 1 + (-0.299 + 1.31i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (-3.77 - 16.5i)T + (-260. + 125. i)T^{2} \) |
| 19 | \( 1 + (19.1 + 15.2i)T + (80.3 + 351. i)T^{2} \) |
| 23 | \( 1 + (12.4 + 15.5i)T + (-117. + 515. i)T^{2} \) |
| 29 | \( 1 + (-10.8 - 22.5i)T + (-524. + 657. i)T^{2} \) |
| 31 | \( 1 + (-3.71 + 1.79i)T + (599. - 751. i)T^{2} \) |
| 37 | \( 1 - 35.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-6.53 + 3.14i)T + (1.04e3 - 1.31e3i)T^{2} \) |
| 47 | \( 1 + (-56.9 + 71.4i)T + (-491. - 2.15e3i)T^{2} \) |
| 53 | \( 1 + (-3.24 - 14.2i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 + (18.1 + 79.3i)T + (-3.13e3 + 1.51e3i)T^{2} \) |
| 61 | \( 1 + (-8.53 + 17.7i)T + (-2.32e3 - 2.90e3i)T^{2} \) |
| 67 | \( 1 + (76.8 - 96.3i)T + (-998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (66.7 + 53.2i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (27.2 + 6.21i)T + (4.80e3 + 2.31e3i)T^{2} \) |
| 79 | \( 1 - 45.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + (19.1 + 9.23i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (-7.38 + 15.3i)T + (-4.93e3 - 6.19e3i)T^{2} \) |
| 97 | \( 1 + (-65.9 - 82.6i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−14.80062691048467245287305015519, −13.48696405380118128427147059527, −12.72309745620365286124003873217, −11.93087047435640445803404158169, −10.37505274347763398754281538656, −9.007646454566575000633525824343, −8.313437824926903191632714966015, −6.30128243186335403619992456246, −2.70834038492346935784496630050, −1.84274628491816555654272653263,
4.35925705958413355876375182540, 5.81804861245939090613513598339, 7.56063141219785579249678987447, 8.908138773840429748758616460340, 9.800843100413294974020575734534, 10.43451295297781903046799782977, 13.75992869343460164500879384622, 14.14493180553955594848786437149, 15.30278438775648300270670703095, 16.32743898261898765635011594214
