L(s) = 1 | + 2.18i·2-s − 2.79·4-s + (−1.5 − 0.866i)5-s + (2.29 − 1.32i)7-s − 1.73i·8-s + (1.89 − 3.28i)10-s + (−3 − 1.73i)11-s + (−1 − 3.46i)13-s + (2.89 + 5.01i)14-s − 1.79·16-s − 17-s + (4.58 − 2.64i)19-s + (4.18 + 2.41i)20-s + (3.79 − 6.56i)22-s + 8.58·23-s + ⋯ |
L(s) = 1 | + 1.54i·2-s − 1.39·4-s + (−0.670 − 0.387i)5-s + (0.866 − 0.499i)7-s − 0.612i·8-s + (0.599 − 1.03i)10-s + (−0.904 − 0.522i)11-s + (−0.277 − 0.960i)13-s + (0.773 + 1.34i)14-s − 0.447·16-s − 0.242·17-s + (1.05 − 0.606i)19-s + (0.936 + 0.540i)20-s + (0.808 − 1.40i)22-s + 1.78·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12280 + 0.187878i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12280 + 0.187878i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.29 + 1.32i)T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 2 | \( 1 - 2.18iT - 2T^{2} \) |
| 5 | \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + (-4.58 + 2.64i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.29 + 3.05i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.02iT - 37T^{2} \) |
| 41 | \( 1 + (-3.08 + 1.77i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.29 - 3.96i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.708 - 0.409i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.08 + 5.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 4.28iT - 59T^{2} \) |
| 61 | \( 1 + (-2.58 - 4.47i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.1 + 7.02i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.87 - 2.23i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.708 + 1.22i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 15.5iT - 89T^{2} \) |
| 97 | \( 1 + (-9.08 - 5.24i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.15273854493555400219369288210, −9.003548173005947204779537982050, −8.125508813466311687285242644371, −7.78959324828657046255901358883, −7.07212731291390002389565129840, −5.85981300806408103781013948009, −5.03150211977101162849352580814, −4.47337038036019021751706779411, −2.91183315498519776141480229563, −0.60727476220761322079540109663,
1.44842099583519400398154705870, 2.53637715545144850508877551089, 3.46330405760525547649910581817, 4.58459826465688741655130203172, 5.29182888887573041405004148736, 7.00233508631424995119038833384, 7.68378962984424336214378756670, 8.865947711921103131378756875200, 9.455618469872235971114996000307, 10.54768205345591201577896735424