| L(s) = 1 | + (1.23 + 0.689i)2-s + (−0.900 + 2.17i)3-s + (1.05 + 1.70i)4-s + (−0.390 + 0.765i)5-s + (−2.61 + 2.06i)6-s + (−0.876 − 3.65i)7-s + (0.124 + 2.82i)8-s + (−1.79 − 1.79i)9-s + (−1.00 + 0.676i)10-s + (1.26 − 1.48i)11-s + (−4.64 + 0.751i)12-s + (1.79 + 2.93i)13-s + (1.43 − 5.11i)14-s + (−1.31 − 1.53i)15-s + (−1.79 + 3.57i)16-s + (−0.302 − 3.84i)17-s + ⋯ |
| L(s) = 1 | + (0.873 + 0.487i)2-s + (−0.519 + 1.25i)3-s + (0.525 + 0.850i)4-s + (−0.174 + 0.342i)5-s + (−1.06 + 0.842i)6-s + (−0.331 − 1.38i)7-s + (0.0440 + 0.999i)8-s + (−0.598 − 0.598i)9-s + (−0.319 + 0.214i)10-s + (0.382 − 0.447i)11-s + (−1.34 + 0.216i)12-s + (0.498 + 0.813i)13-s + (0.383 − 1.36i)14-s + (−0.339 − 0.397i)15-s + (−0.448 + 0.893i)16-s + (−0.0733 − 0.932i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.888249 + 1.22736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.888249 + 1.22736i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 - 0.689i)T \) |
| 41 | \( 1 + (5.03 - 3.96i)T \) |
| good | 3 | \( 1 + (0.900 - 2.17i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.390 - 0.765i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (0.876 + 3.65i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-1.26 + 1.48i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.79 - 2.93i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (0.302 + 3.84i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-5.10 - 3.12i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (1.43 + 1.03i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.535 + 6.80i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-0.437 + 1.34i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.67 + 8.24i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (1.17 + 7.43i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.70 + 7.11i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (3.84 + 0.302i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (7.28 - 10.0i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.26 - 8.01i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-7.01 + 5.99i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (2.24 + 1.91i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-8.30 + 8.30i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.23 - 0.512i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 13.0iT - 83T^{2} \) |
| 89 | \( 1 + (6.69 - 1.60i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (8.34 - 7.12i)T + (15.1 - 95.8i)T^{2} \) |
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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
−13.63543023191680452110072300199, −11.94123135576619460706255486422, −11.21248267582320693624866017002, −10.37177962655609775778356695593, −9.248569770279766457350802787305, −7.59888662981607428280333028344, −6.64273013980910513219309689585, −5.38661277042405158915079359203, −4.15376833821080000514684066906, −3.53463043167813675368540304548,
1.48738573938578139798780905436, 3.07792199205739655621176198965, 5.04366458257841770342019123891, 6.02086372755148338190144095980, 6.83913977013820209910240964939, 8.306898387021122826815474636778, 9.626079048769327859940677768008, 11.04679295691730494245537601703, 12.01432085920075530876751387469, 12.49780559204483661842135631130
