| L(s) = 1 | + (−0.462 − 1.33i)2-s + (−2.24 − 0.930i)3-s + (−1.57 + 1.23i)4-s + (−1.19 + 0.607i)5-s + (−0.204 + 3.43i)6-s + (0.861 + 1.40i)7-s + (2.37 + 1.52i)8-s + (2.06 + 2.06i)9-s + (1.36 + 1.31i)10-s + (−1.90 − 0.150i)11-s + (4.68 − 1.31i)12-s + (0.816 + 3.40i)13-s + (1.48 − 1.80i)14-s + (3.24 − 0.255i)15-s + (0.942 − 3.88i)16-s + (−4.46 + 3.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.327 − 0.944i)2-s + (−1.29 − 0.537i)3-s + (−0.786 + 0.618i)4-s + (−0.533 + 0.271i)5-s + (−0.0834 + 1.40i)6-s + (0.325 + 0.531i)7-s + (0.841 + 0.540i)8-s + (0.686 + 0.686i)9-s + (0.431 + 0.415i)10-s + (−0.575 − 0.0452i)11-s + (1.35 − 0.379i)12-s + (0.226 + 0.943i)13-s + (0.395 − 0.481i)14-s + (0.838 − 0.0659i)15-s + (0.235 − 0.971i)16-s + (−1.08 + 0.924i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.187913 + 0.128200i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.187913 + 0.128200i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.462 + 1.33i)T \) |
| 41 | \( 1 + (-6.30 - 1.14i)T \) |
| good | 3 | \( 1 + (2.24 + 0.930i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (1.19 - 0.607i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.861 - 1.40i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (1.90 + 0.150i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (-0.816 - 3.40i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (4.46 - 3.81i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (1.57 + 0.378i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (1.05 - 0.763i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (7.58 + 6.47i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-2.13 - 6.57i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.802 + 2.46i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (10.9 + 1.72i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-2.02 + 3.30i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (2.84 - 3.32i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (3.96 + 5.45i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.73 + 1.38i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-1.15 - 14.6i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-0.561 + 7.13i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-1.44 + 1.44i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.46 - 5.93i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 0.232iT - 83T^{2} \) |
| 89 | \( 1 + (-3.81 + 2.33i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (-0.857 - 10.8i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−12.70832071109347234285110818701, −11.68266046461409409199793604338, −11.33818335841756809387420619828, −10.47714581716336043805264103126, −9.045610636343553402815216727245, −7.942688054334753641424394949641, −6.68122398095947102605858872287, −5.35876504907780334352485620189, −4.02042934070244337554842845451, −1.96908502029794433833844904219,
0.27649383377418900189954547710, 4.26208587745623015587951699750, 5.09413395451266170250087433190, 6.11756157826009404632735760213, 7.36948101593232296873636394953, 8.332837179541928997965238228189, 9.703847347109394011723672945554, 10.69364821450164861308842458223, 11.32277161028021732439722220910, 12.70371388586261557188406236477
