L(s) = 1 | + (1.29 + 0.561i)2-s + (−1.27 + 1.16i)3-s + (1.36 + 1.45i)4-s + (−0.432 + 0.249i)5-s + (−2.31 + 0.798i)6-s + (−0.261 − 2.63i)7-s + (0.956 + 2.66i)8-s + (0.267 − 2.98i)9-s + (−0.701 + 0.0810i)10-s + (0.695 − 1.20i)11-s + (−3.45 − 0.264i)12-s + 2.75·13-s + (1.13 − 3.56i)14-s + (0.260 − 0.824i)15-s + (−0.255 + 3.99i)16-s + (−5.04 − 2.91i)17-s + ⋯ |
L(s) = 1 | + (0.917 + 0.397i)2-s + (−0.737 + 0.674i)3-s + (0.684 + 0.729i)4-s + (−0.193 + 0.111i)5-s + (−0.945 + 0.326i)6-s + (−0.0989 − 0.995i)7-s + (0.338 + 0.941i)8-s + (0.0890 − 0.996i)9-s + (−0.221 + 0.0256i)10-s + (0.209 − 0.363i)11-s + (−0.997 − 0.0763i)12-s + 0.762·13-s + (0.304 − 0.952i)14-s + (0.0673 − 0.212i)15-s + (−0.0637 + 0.997i)16-s + (−1.22 − 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08583 + 0.597802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08583 + 0.597802i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 - 0.561i)T \) |
| 3 | \( 1 + (1.27 - 1.16i)T \) |
| 7 | \( 1 + (0.261 + 2.63i)T \) |
good | 5 | \( 1 + (0.432 - 0.249i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.695 + 1.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + (5.04 + 2.91i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.14 - 1.24i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.53 + 4.38i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.32iT - 29T^{2} \) |
| 31 | \( 1 + (-5.98 - 3.45i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.67 + 6.35i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.74iT - 41T^{2} \) |
| 43 | \( 1 - 3.52iT - 43T^{2} \) |
| 47 | \( 1 + (-2.53 - 4.38i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.54 - 2.62i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.57 + 2.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.45 - 4.25i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.30 + 5.37i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.54T + 71T^{2} \) |
| 73 | \( 1 + (-4.98 + 8.63i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.84 + 1.06i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + (-2.12 + 1.22i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.526T + 97T^{2} \) |
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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
−14.42277102586069767685273002952, −13.52020353930271227580821579015, −12.34219228820005812733581127288, −11.17757420039086540525515757763, −10.57131168799751132294792247462, −8.780339467099287732162246553368, −7.10321160765823228613685444763, −6.14456189761037719422754096065, −4.64858799110447758300727586014, −3.63063294412895302163054606797,
2.13605756298361284011251679922, 4.35925652640089098183772836868, 5.83235948350080755315369453418, 6.60861210262441450873925886296, 8.324735873989209646366027576015, 10.07676124749968817030281171541, 11.41069082486158563935731257528, 11.92002555992498771768224117454, 12.99933315499747855590973195584, 13.69033187964255442572687465889