| L(s) = 1 | + (−0.887 + 1.10i)2-s + (−0.435 + 2.18i)3-s + (−0.423 − 1.95i)4-s + (0.649 + 0.434i)5-s + (−2.02 − 2.42i)6-s + (−3.64 + 1.50i)7-s + (2.52 + 1.26i)8-s + (−1.82 − 0.757i)9-s + (−1.05 + 0.329i)10-s + (5.80 − 1.15i)11-s + (4.46 − 0.0758i)12-s + (2.03 − 1.35i)13-s + (1.57 − 5.35i)14-s + (−1.23 + 1.23i)15-s + (−3.64 + 1.65i)16-s + (0.960 + 0.960i)17-s + ⋯ |
| L(s) = 1 | + (−0.627 + 0.778i)2-s + (−0.251 + 1.26i)3-s + (−0.211 − 0.977i)4-s + (0.290 + 0.194i)5-s + (−0.825 − 0.988i)6-s + (−1.37 + 0.570i)7-s + (0.893 + 0.448i)8-s + (−0.609 − 0.252i)9-s + (−0.333 + 0.104i)10-s + (1.74 − 0.347i)11-s + (1.28 − 0.0219i)12-s + (0.564 − 0.377i)13-s + (0.420 − 1.43i)14-s + (−0.318 + 0.318i)15-s + (−0.910 + 0.413i)16-s + (0.232 + 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.304674 + 0.539038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.304674 + 0.539038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.887 - 1.10i)T \) |
| good | 3 | \( 1 + (0.435 - 2.18i)T + (-2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (-0.649 - 0.434i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (3.64 - 1.50i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-5.80 + 1.15i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.03 + 1.35i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-0.960 - 0.960i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.435 - 0.652i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.421 + 1.01i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (1.43 + 0.286i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 6.88iT - 31T^{2} \) |
| 37 | \( 1 + (1.07 - 1.61i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (2.79 - 6.74i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (1.20 + 6.04i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (6.30 + 6.30i)T + 47iT^{2} \) |
| 53 | \( 1 + (-10.2 + 2.04i)T + (48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-4.21 - 2.81i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (2.08 - 10.4i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-1.42 + 7.14i)T + (-61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-2.49 + 1.03i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (1.90 + 0.789i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (6.20 - 6.20i)T - 79iT^{2} \) |
| 83 | \( 1 + (-1.13 - 1.70i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-1.15 - 2.79i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 13.1iT - 97T^{2} \) |
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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
−15.47389604047537353663873564829, −14.73401527572900342950818391143, −13.41820990127432594728565239485, −11.64667505796590253088462172548, −10.24856161269632252016820548830, −9.595793364740027518328428969333, −8.700726141998114554998317487859, −6.58438098416527644214149130055, −5.74789369202124290877326057919, −3.86189465559946838279568488304,
1.38085145079956726753184602736, 3.66322805352965682166040356489, 6.50473209801363235514462664862, 7.22196754139112172207649851938, 8.979878052050208433192331490493, 9.854562059169724225512291060693, 11.42779257744083638490385947698, 12.35831424115906196394069985426, 13.13888515806521807537461584529, 13.98679308671897055376687673049
