L(s) = 1 | + (−0.927 + 1.06i)2-s + (−0.280 − 1.98i)4-s + 2i·5-s − 0.936·7-s + (2.37 + 1.53i)8-s + (−2.13 − 1.85i)10-s + (3.09 − 1.19i)11-s − 4.27·13-s + (0.868 − 0.999i)14-s + (−3.84 + 1.11i)16-s + 3.33i·17-s + 2.89i·19-s + (3.96 − 0.561i)20-s + (−1.58 + 4.41i)22-s + 5.12i·23-s + ⋯ |
L(s) = 1 | + (−0.655 + 0.755i)2-s + (−0.140 − 0.990i)4-s + 0.894i·5-s − 0.353·7-s + (0.839 + 0.543i)8-s + (−0.675 − 0.586i)10-s + (0.932 − 0.361i)11-s − 1.18·13-s + (0.232 − 0.267i)14-s + (−0.960 + 0.277i)16-s + 0.808i·17-s + 0.664i·19-s + (0.885 − 0.125i)20-s + (−0.338 + 0.941i)22-s + 1.06i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0666891 + 0.651177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0666891 + 0.651177i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.927 - 1.06i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-3.09 + 1.19i)T \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 0.936T + 7T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 - 3.33iT - 17T^{2} \) |
| 19 | \( 1 - 2.89iT - 19T^{2} \) |
| 23 | \( 1 - 5.12iT - 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 6.18iT - 37T^{2} \) |
| 41 | \( 1 - 1.46iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 8.24iT - 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 + 9.06T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 4.24iT - 71T^{2} \) |
| 73 | \( 1 + 13.2iT - 73T^{2} \) |
| 79 | \( 1 - 3.86T + 79T^{2} \) |
| 83 | \( 1 - 2.39iT - 83T^{2} \) |
| 89 | \( 1 + 3.47T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−10.56875330786496217706219951234, −9.564339703586645104111105950612, −9.200214174459553813162840812371, −7.84427910321719446960875598417, −7.36312022554439330801690983110, −6.34388577686130888031429661458, −5.82843806233462428421455790943, −4.45529826220840321935330031681, −3.19781763988487174399186630851, −1.67667132149575047671009735170,
0.40512564365465397620767700593, 1.87797013250409026934592418431, 3.09764281491793082832232871989, 4.36947793138191347084963996881, 5.06419976555468779549317807869, 6.71676066503172070880715796781, 7.37642917735282081853799746934, 8.522081711533516802982122154160, 9.185208312994311233975104647393, 9.708758232244516794254409029372