| L(s) = 1 | + (−1.00 + 0.992i)2-s + (1.26 + 0.622i)3-s + (0.0313 − 1.99i)4-s + (0.932 − 2.74i)5-s + (−1.89 + 0.625i)6-s + (−2.32 − 1.26i)7-s + (1.95 + 2.04i)8-s + (−0.619 − 0.807i)9-s + (1.78 + 3.69i)10-s + (−3.25 − 0.213i)11-s + (1.28 − 2.50i)12-s + (1.92 + 0.383i)13-s + (3.59 − 1.03i)14-s + (2.88 − 2.88i)15-s + (−3.99 − 0.125i)16-s + (1.01 − 3.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.712 + 0.701i)2-s + (0.728 + 0.359i)3-s + (0.0156 − 0.999i)4-s + (0.417 − 1.22i)5-s + (−0.771 + 0.255i)6-s + (−0.879 − 0.476i)7-s + (0.690 + 0.723i)8-s + (−0.206 − 0.269i)9-s + (0.564 + 1.16i)10-s + (−0.980 − 0.0642i)11-s + (0.370 − 0.723i)12-s + (0.535 + 0.106i)13-s + (0.960 − 0.277i)14-s + (0.745 − 0.745i)15-s + (−0.999 − 0.0313i)16-s + (0.245 − 0.916i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.799012 - 0.490977i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.799012 - 0.490977i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.00 - 0.992i)T \) |
| 7 | \( 1 + (2.32 + 1.26i)T \) |
| good | 3 | \( 1 + (-1.26 - 0.622i)T + (1.82 + 2.38i)T^{2} \) |
| 5 | \( 1 + (-0.932 + 2.74i)T + (-3.96 - 3.04i)T^{2} \) |
| 11 | \( 1 + (3.25 + 0.213i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (-1.92 - 0.383i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (-1.01 + 3.77i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (4.66 + 4.09i)T + (2.47 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.63 - 2.02i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-1.79 - 2.68i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-2.95 + 1.70i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.72 - 1.60i)T + (29.3 + 22.5i)T^{2} \) |
| 41 | \( 1 + (-1.87 - 0.778i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (5.22 + 3.48i)T + (16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (-9.22 + 2.47i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-7.23 - 0.474i)T + (52.5 + 6.91i)T^{2} \) |
| 59 | \( 1 + (-6.28 - 7.16i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.190 + 2.90i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-1.23 - 0.611i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-5.85 - 14.1i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (3.84 - 0.506i)T + (70.5 - 18.8i)T^{2} \) |
| 79 | \( 1 + (3.22 + 12.0i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 6.16i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (-1.16 + 8.86i)T + (-85.9 - 23.0i)T^{2} \) |
| 97 | \( 1 + 8.53iT - 97T^{2} \) |
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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
−10.42671602591917651413689316707, −9.781505635922805094812296016129, −8.942422655633459847756002748301, −8.559142947108410281043593197602, −7.43570720049544299996885710167, −6.30403836208525711012127450606, −5.34009869019231315500690136634, −4.22949377316302282932551037664, −2.59568635005871309790107090499, −0.65076781567696486640733345061,
2.18042508533463149644428556207, 2.75840155666996996464877274434, 3.79493165641863443020276974587, 5.91175222430629217852742825640, 6.76960460968323097626009487449, 7.985304180352287705265969724196, 8.414679101174134619256997749224, 9.619460410003501986549442705235, 10.46292284713908079614725323987, 10.79497228981963165818970928177
