L(s) = 1 | + (−0.445 + 0.184i)3-s + (0.565 − 1.36i)5-s + (−0.135 − 0.135i)7-s + (−1.95 + 1.95i)9-s + (−3.12 − 1.29i)11-s + (0.951 + 2.29i)13-s + 0.713i·15-s − 3.11i·17-s + (−2.48 − 5.99i)19-s + (0.0851 + 0.0352i)21-s + (−5.18 + 5.18i)23-s + (1.98 + 1.98i)25-s + (1.06 − 2.57i)27-s + (−4.33 + 1.79i)29-s − 7.44·31-s + ⋯ |
L(s) = 1 | + (−0.257 + 0.106i)3-s + (0.253 − 0.610i)5-s + (−0.0510 − 0.0510i)7-s + (−0.652 + 0.652i)9-s + (−0.941 − 0.390i)11-s + (0.263 + 0.637i)13-s + 0.184i·15-s − 0.754i·17-s + (−0.569 − 1.37i)19-s + (0.0185 + 0.00769i)21-s + (−1.08 + 1.08i)23-s + (0.397 + 0.397i)25-s + (0.204 − 0.494i)27-s + (−0.804 + 0.333i)29-s − 1.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1354669800\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1354669800\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.445 - 0.184i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.565 + 1.36i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (0.135 + 0.135i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.12 + 1.29i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 2.29i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 3.11iT - 17T^{2} \) |
| 19 | \( 1 + (2.48 + 5.99i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (5.18 - 5.18i)T - 23iT^{2} \) |
| 29 | \( 1 + (4.33 - 1.79i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 + (-3.49 + 8.44i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (4.27 - 4.27i)T - 41iT^{2} \) |
| 43 | \( 1 + (4.33 + 1.79i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 12.0iT - 47T^{2} \) |
| 53 | \( 1 + (-3.42 - 1.41i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.16 + 2.81i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (8.72 - 3.61i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (7.15 - 2.96i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-2.86 - 2.86i)T + 71iT^{2} \) |
| 73 | \( 1 + (2.49 - 2.49i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (5.42 + 13.0i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (4.96 + 4.96i)T + 89iT^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.334168217070641018541319485045, −8.896034778656139058878187240226, −7.87895402368237222873739798886, −7.10423901958853413902599482212, −5.82553204973867827867751038575, −5.29724907009095610581630321946, −4.41100138938749220833980981622, −3.04893926218310891389754918664, −1.89328543413997739222236572975, −0.05813472493575223112654162793,
1.94693064198729849934727754520, 3.07575643778026386981282853884, 4.06646744959736533029865954147, 5.46483525868767946774859670864, 6.06087541562822154386703529001, 6.82477198709847269004790987171, 8.020782744995986493162429710936, 8.475414441184735191053176793264, 9.743973059796266487154370375451, 10.44127989950925984581700459416