L(s) = 1 | + (0.415 − 0.909i)2-s + (1.59 − 1.02i)3-s + (−0.654 − 0.755i)4-s + (0.142 − 0.989i)5-s + (−0.270 − 1.87i)6-s + (0.123 − 0.269i)7-s + (−0.959 + 0.281i)8-s + (0.249 − 0.545i)9-s + (−0.841 − 0.540i)10-s + (0.287 − 1.99i)11-s + (−1.82 − 0.534i)12-s + (−3.20 − 0.941i)13-s + (−0.194 − 0.224i)14-s + (−0.788 − 1.72i)15-s + (−0.142 + 0.989i)16-s + (3.77 − 4.35i)17-s + ⋯ |
L(s) = 1 | + (0.293 − 0.643i)2-s + (0.921 − 0.592i)3-s + (−0.327 − 0.377i)4-s + (0.0636 − 0.442i)5-s + (−0.110 − 0.766i)6-s + (0.0465 − 0.101i)7-s + (−0.339 + 0.0996i)8-s + (0.0831 − 0.181i)9-s + (−0.266 − 0.170i)10-s + (0.0865 − 0.602i)11-s + (−0.525 − 0.154i)12-s + (−0.889 − 0.261i)13-s + (−0.0519 − 0.0599i)14-s + (−0.203 − 0.445i)15-s + (−0.0355 + 0.247i)16-s + (0.914 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.885570 - 1.94665i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.885570 - 1.94665i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (-8.17 + 0.414i)T \) |
good | 3 | \( 1 + (-1.59 + 1.02i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (-0.123 + 0.269i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.287 + 1.99i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.20 + 0.941i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-3.77 + 4.35i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.64 + 3.59i)T + (-12.4 + 14.3i)T^{2} \) |
| 23 | \( 1 + (-0.266 + 0.171i)T + (9.55 - 20.9i)T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 + (-3.78 + 1.11i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 - 1.50T + 37T^{2} \) |
| 41 | \( 1 + (8.03 - 9.27i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (4.25 - 4.91i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-8.64 + 5.55i)T + (19.5 - 42.7i)T^{2} \) |
| 53 | \( 1 + (-6.53 - 7.53i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (3.65 - 1.07i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (1.23 + 8.58i)T + (-58.5 + 17.1i)T^{2} \) |
| 71 | \( 1 + (-0.175 - 0.202i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.377 - 2.62i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-5.09 - 1.49i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (1.83 - 12.7i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (4.93 + 3.16i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−10.10312867786265161129126061315, −9.343806018087524489054478529279, −8.483841545610094364643911250073, −7.76863344126418623724248054816, −6.76043375258708784717135863894, −5.41404925158600347932927619759, −4.58257194804074391063841882237, −3.13691793589581136198922704457, −2.46752759779616535557427720781, −0.981443399671304438877518888796,
2.23804867725653168882368041079, 3.45428920355700175226140190431, 4.21112655480545168435302612862, 5.37123418835350496956862175050, 6.43703236466609377756644334431, 7.36665220982391012734453858157, 8.252854857873946602912172739998, 8.937868048473315754008786537279, 10.03507172348074530948750718356, 10.30871879351643918384396436323