| L(s) = 1 | + (1.26 − 0.633i)2-s + (0.152 + 0.769i)3-s + (1.19 − 1.60i)4-s + (−2.78 + 1.86i)5-s + (0.680 + 0.875i)6-s + (−3.13 − 1.29i)7-s + (0.497 − 2.78i)8-s + (2.20 − 0.912i)9-s + (−2.34 + 4.12i)10-s + (−1.79 − 0.356i)11-s + (1.41 + 0.675i)12-s + (4.34 + 2.90i)13-s + (−4.78 + 0.345i)14-s + (−1.85 − 1.85i)15-s + (−1.13 − 3.83i)16-s + (−1.16 + 1.16i)17-s + ⋯ |
| L(s) = 1 | + (0.893 − 0.448i)2-s + (0.0883 + 0.443i)3-s + (0.598 − 0.801i)4-s + (−1.24 + 0.832i)5-s + (0.277 + 0.357i)6-s + (−1.18 − 0.491i)7-s + (0.175 − 0.984i)8-s + (0.734 − 0.304i)9-s + (−0.741 + 1.30i)10-s + (−0.540 − 0.107i)11-s + (0.408 + 0.194i)12-s + (1.20 + 0.805i)13-s + (−1.27 + 0.0922i)14-s + (−0.479 − 0.479i)15-s + (−0.283 − 0.958i)16-s + (−0.281 + 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18081 - 0.151708i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18081 - 0.151708i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 + 0.633i)T \) |
| good | 3 | \( 1 + (-0.152 - 0.769i)T + (-2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (2.78 - 1.86i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (3.13 + 1.29i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (1.79 + 0.356i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-4.34 - 2.90i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (1.16 - 1.16i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.00265 - 0.00396i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.78 - 4.32i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (1.70 - 0.338i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + 9.42iT - 31T^{2} \) |
| 37 | \( 1 + (1.47 + 2.20i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-0.497 - 1.20i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.104 + 0.523i)T + (-39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (0.378 - 0.378i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.63 + 0.723i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (11.1 - 7.47i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (1.29 + 6.52i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-1.46 - 7.38i)T + (-61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-5.03 - 2.08i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (4.53 - 1.87i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-11.3 - 11.3i)T + 79iT^{2} \) |
| 83 | \( 1 + (-3.35 + 5.01i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-4.25 + 10.2i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−15.08823032861232943619115887364, −13.66167986918675816095134068329, −12.79409957292559220944762617049, −11.44116379516242037053973165557, −10.68392858829128522292397449169, −9.523484386097015366513387811735, −7.36152629887266824751923336212, −6.33995298684375690895970375962, −4.09886128698648274356674370196, −3.41754238373031138205806664179,
3.33428409105384288303172876887, 4.83969789436787060691265920493, 6.46395609159756665959118894768, 7.72422373979125074338857016346, 8.708704080990056840878943809855, 10.77782737880719561800238930359, 12.31529607456911224021933841677, 12.71258563467290661346691794440, 13.56349544002730324871614894709, 15.38201159245546629599799583502
