L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s − 5-s − 0.999·6-s + (−3.72 + 2.70i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (0.659 − 0.479i)11-s + (−0.309 − 0.951i)12-s + (1.93 − 5.96i)13-s + (−3.72 − 2.70i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−4.62 − 3.36i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.178 + 0.549i)3-s + (−0.404 + 0.293i)4-s − 0.447·5-s − 0.408·6-s + (−1.40 + 1.02i)7-s + (−0.286 − 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0977 − 0.300i)10-s + (0.198 − 0.144i)11-s + (−0.0892 − 0.274i)12-s + (0.537 − 1.65i)13-s + (−0.996 − 0.723i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (−1.12 − 0.815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.297321 - 0.162199i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.297321 - 0.162199i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + (-4.44 - 3.35i)T \) |
good | 7 | \( 1 + (3.72 - 2.70i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.659 + 0.479i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.93 + 5.96i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.62 + 3.36i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.0338 - 0.104i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.09 - 2.24i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.434 - 1.33i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + (-0.403 - 1.24i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (0.288 + 0.888i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-3.21 + 9.88i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (8.68 + 6.30i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.39 + 4.30i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 + 8.54T + 67T^{2} \) |
| 71 | \( 1 + (11.6 + 8.49i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.60 - 6.25i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-7.39 - 5.37i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.05 + 9.39i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-3.64 + 2.64i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.72 + 2.70i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.816142454205902996366848302866, −8.914088679983430768557754281414, −8.491260083971764805144563197922, −7.20971565151980644903597452640, −6.39695757030608690557498143555, −5.62321649463382989360289095949, −4.83439086206750038649471227654, −3.45537778082157512335237085381, −2.95516506296401547095241702400, −0.15809276048580369159308930280,
1.37418736817007008142737809737, 2.79577421141936915166877954651, 3.97641618099919568825547489210, 4.45135149692886303952602930088, 6.25576011179128882934896364208, 6.58845657045629284596974240159, 7.50413231797632017495658653250, 8.798176659180239635156247304407, 9.349748361172745682844068373182, 10.44922522045925569751361242494