L(s) = 1 | + (1.03 − 0.968i)2-s + (1.63 − 0.565i)3-s + (0.123 − 1.99i)4-s + (−0.289 − 0.329i)5-s + (1.13 − 2.16i)6-s + (−0.132 − 0.0173i)7-s + (−1.80 − 2.17i)8-s + (2.35 − 1.85i)9-s + (−0.617 − 0.0597i)10-s + (−2.45 − 4.98i)11-s + (−0.926 − 3.33i)12-s + (1.43 + 4.22i)13-s + (−0.153 + 0.110i)14-s + (−0.660 − 0.376i)15-s + (−3.96 − 0.494i)16-s + (0.861 + 0.861i)17-s + ⋯ |
L(s) = 1 | + (0.728 − 0.684i)2-s + (0.945 − 0.326i)3-s + (0.0618 − 0.998i)4-s + (−0.129 − 0.147i)5-s + (0.464 − 0.885i)6-s + (−0.0499 − 0.00657i)7-s + (−0.638 − 0.769i)8-s + (0.786 − 0.617i)9-s + (−0.195 − 0.0188i)10-s + (−0.740 − 1.50i)11-s + (−0.267 − 0.963i)12-s + (0.398 + 1.17i)13-s + (−0.0409 + 0.0294i)14-s + (−0.170 − 0.0971i)15-s + (−0.992 − 0.123i)16-s + (0.208 + 0.208i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60220 - 2.21086i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60220 - 2.21086i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.03 + 0.968i)T \) |
| 3 | \( 1 + (-1.63 + 0.565i)T \) |
good | 5 | \( 1 + (0.289 + 0.329i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (0.132 + 0.0173i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (2.45 + 4.98i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-1.43 - 4.22i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-0.861 - 0.861i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.11 - 5.60i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.478 - 3.63i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-5.16 + 0.338i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-6.68 + 3.86i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.738 - 3.71i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.487 - 3.70i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (3.01 - 1.48i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (1.60 - 0.429i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.07 + 6.09i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-5.53 - 6.31i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.930 - 14.1i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (5.53 + 2.72i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (4.00 + 9.67i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (2.92 - 7.05i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.29 - 8.57i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-1.38 - 1.21i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-2.54 + 1.05i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-11.1 - 6.41i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.43218195150917059375944284303, −9.759819747994830500749600167543, −8.616583831569384006631049397842, −8.078242460141441264313806252909, −6.60792238893516609995972359787, −5.89379256109075928067776373899, −4.46814445144355253683858042940, −3.56416463349735287632649920315, −2.62471273814118479162862428611, −1.26000445869096978983945324898,
2.46411268693879104250202156753, 3.26529997568170622201382345269, 4.57688046821167995966082993752, 5.12430282490945922392181837294, 6.66067516460470842041303856435, 7.42753939121692246334488785437, 8.155933841102874353033450349882, 9.001826925051078892936293300580, 10.08017277510493792803129134795, 10.85805003606879987876944901562