| L(s) = 1 | + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.326 − 1.85i)5-s + (0.598 − 3.39i)7-s + (0.500 + 0.866i)8-s + (0.939 − 1.62i)10-s + (−1.40 − 2.43i)11-s + (−2.65 − 2.23i)13-s + (1.72 − 2.98i)14-s + (0.173 + 0.984i)16-s + (2.37 − 1.99i)17-s + (−6.99 + 2.54i)19-s + (1.43 − 1.20i)20-s + (−0.487 − 2.76i)22-s + (−0.321 + 0.557i)23-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.383 + 0.321i)4-s + (0.145 − 0.827i)5-s + (0.226 − 1.28i)7-s + (0.176 + 0.306i)8-s + (0.297 − 0.514i)10-s + (−0.423 − 0.733i)11-s + (−0.737 − 0.618i)13-s + (0.460 − 0.797i)14-s + (0.0434 + 0.246i)16-s + (0.575 − 0.483i)17-s + (−1.60 + 0.584i)19-s + (0.321 − 0.270i)20-s + (−0.104 − 0.589i)22-s + (−0.0670 + 0.116i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79647 - 1.07440i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79647 - 1.07440i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (2.56 + 5.51i)T \) |
| good | 5 | \( 1 + (-0.326 + 1.85i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.598 + 3.39i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.40 + 2.43i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.65 + 2.23i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.37 + 1.99i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (6.99 - 2.54i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (0.321 - 0.557i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.08 - 1.87i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 9.90T + 31T^{2} \) |
| 41 | \( 1 + (-8.13 - 6.82i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 - 8.30T + 43T^{2} \) |
| 47 | \( 1 + (3.92 - 6.80i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.839 + 4.76i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.0961 + 0.545i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.37 - 4.50i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.0366 + 0.207i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.75 + 1.72i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + (0.330 - 1.87i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.21 - 4.37i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.68 - 9.54i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (8.52 - 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.52943980849200676634922806287, −9.595407496430149986571594203930, −8.297600622170646933677024240203, −7.82377784057662982368150530474, −6.75494606884730766238172184616, −5.70898736919676887241487597725, −4.79534756249519251738542238534, −4.03854130950940173508727146420, −2.72743806887506311090510734989, −0.921723552521936472586031955499,
2.21058255351989845883074226778, 2.66604023663586399225547126700, 4.24789576557309715947646382507, 5.12232689665803410987379519788, 6.20222657305747209925325754915, 6.84135430734613072442075725207, 8.017040553415288444708529111963, 9.012738321905132961231988489282, 10.05361796342063134703927182945, 10.66712093790581356292172406501
