L(s) = 1 | + (−2.70 − 1.12i)3-s + (0.151 + 0.366i)5-s + (−3.06 + 3.06i)7-s + (3.95 + 3.95i)9-s + (3.66 − 1.51i)11-s + (0.780 − 1.88i)13-s − 1.16i·15-s + 4.54i·17-s + (−0.221 + 0.534i)19-s + (11.7 − 4.86i)21-s + (−4.41 − 4.41i)23-s + (3.42 − 3.42i)25-s + (−2.91 − 7.03i)27-s + (−4.74 − 1.96i)29-s − 0.0539·31-s + ⋯ |
L(s) = 1 | + (−1.56 − 0.647i)3-s + (0.0678 + 0.163i)5-s + (−1.15 + 1.15i)7-s + (1.31 + 1.31i)9-s + (1.10 − 0.457i)11-s + (0.216 − 0.522i)13-s − 0.299i·15-s + 1.10i·17-s + (−0.0508 + 0.122i)19-s + (2.56 − 1.06i)21-s + (−0.921 − 0.921i)23-s + (0.684 − 0.684i)25-s + (−0.560 − 1.35i)27-s + (−0.881 − 0.365i)29-s − 0.00969·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2030031415\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2030031415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (2.70 + 1.12i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.151 - 0.366i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (3.06 - 3.06i)T - 7iT^{2} \) |
| 11 | \( 1 + (-3.66 + 1.51i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.780 + 1.88i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 4.54iT - 17T^{2} \) |
| 19 | \( 1 + (0.221 - 0.534i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (4.41 + 4.41i)T + 23iT^{2} \) |
| 29 | \( 1 + (4.74 + 1.96i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 0.0539T + 31T^{2} \) |
| 37 | \( 1 + (0.330 + 0.798i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.621 - 0.621i)T + 41iT^{2} \) |
| 43 | \( 1 + (2.06 - 0.857i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 9.44iT - 47T^{2} \) |
| 53 | \( 1 + (10.0 - 4.16i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (2.97 + 7.17i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (9.72 + 4.02i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (7.53 + 3.12i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-2.99 + 2.99i)T - 71iT^{2} \) |
| 73 | \( 1 + (2.91 + 2.91i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.74iT - 79T^{2} \) |
| 83 | \( 1 + (-1.36 + 3.30i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-2.38 + 2.38i)T - 89iT^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−9.704119692027211458925984525911, −8.810286541208160524564153139523, −7.81055076937984966379725173421, −6.46537090679479742324433817211, −6.25220336122974423741372580948, −5.75314840690221507046690149781, −4.45299334305441492941356016382, −3.14991563124120784159803531247, −1.69390748381614658943377926018, −0.12667187062050353386012423966,
1.24609081490148484888195392313, 3.52573331918102550953167604468, 4.18485480575017594973871937526, 5.09948137859263616350064324901, 6.08426410512661510947528239250, 6.82278115220192681735697288230, 7.31153147479860578851076420762, 9.152591310549899140967044997494, 9.621573384475259837215641443305, 10.27625963068449936385978261609