L(s) = 1 | + (−1.58 + 1.22i)2-s + (1.01 − 3.86i)4-s − 8.58i·5-s + 1.97i·7-s + (3.11 + 7.36i)8-s + (10.4 + 13.5i)10-s + 19.5·11-s + 18.4i·13-s + (−2.41 − 3.12i)14-s + (−13.9 − 7.86i)16-s + 22.2i·17-s + (−18.8 − 2.33i)19-s + (−33.1 − 8.72i)20-s + (−30.9 + 23.8i)22-s + 15.3·23-s + ⋯ |
L(s) = 1 | + (−0.791 + 0.610i)2-s + (0.254 − 0.967i)4-s − 1.71i·5-s + 0.282i·7-s + (0.389 + 0.921i)8-s + (1.04 + 1.35i)10-s + 1.77·11-s + 1.41i·13-s + (−0.172 − 0.223i)14-s + (−0.870 − 0.491i)16-s + 1.30i·17-s + (−0.992 − 0.122i)19-s + (−1.65 − 0.436i)20-s + (−1.40 + 1.08i)22-s + 0.665·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.357220284\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357220284\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.58 - 1.22i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (18.8 + 2.33i)T \) |
good | 5 | \( 1 + 8.58iT - 25T^{2} \) |
| 7 | \( 1 - 1.97iT - 49T^{2} \) |
| 11 | \( 1 - 19.5T + 121T^{2} \) |
| 13 | \( 1 - 18.4iT - 169T^{2} \) |
| 17 | \( 1 - 22.2iT - 289T^{2} \) |
| 23 | \( 1 - 15.3T + 529T^{2} \) |
| 29 | \( 1 - 38.6T + 841T^{2} \) |
| 31 | \( 1 - 45.2T + 961T^{2} \) |
| 37 | \( 1 - 9.92iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 25.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 6.62T + 2.20e3T^{2} \) |
| 53 | \( 1 + 68.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 83.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 14.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 43.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 84.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 11.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 58.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 30.6iT - 9.40e3T^{2} \) |
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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
−9.900239628921217179367696527210, −9.077238953621674353031277361175, −8.782593244586457034539005894924, −8.046004513205382628658529307183, −6.53477879091534283387624029497, −6.22209327844879734790398303828, −4.75531945032698483343069821221, −4.23737093793701942103076670686, −1.81917904471599038169828912957, −1.01958673975196651403667270564,
0.866422975781856968221093352743, 2.56753606366817702916433636216, 3.23544982555357658139777410171, 4.30937233992972043515772555500, 6.23206533100489447122279478956, 6.86467884290397236066626339350, 7.58155549374089292188902126551, 8.629113117714386225670401672653, 9.606879770204624921702385093262, 10.35947305438729592897955422731