| L(s) = 1 | + (−0.829 − 1.62i)2-s + (0.391 + 1.68i)3-s + (−0.785 + 1.08i)4-s + (−0.189 − 1.19i)5-s + (2.42 − 2.03i)6-s + (−0.760 + 0.649i)7-s + (−1.19 − 0.189i)8-s + (−2.69 + 1.32i)9-s + (−1.78 + 1.29i)10-s + (2.21 + 1.35i)11-s + (−2.13 − 0.902i)12-s + (−0.322 + 0.0254i)13-s + (1.68 + 0.699i)14-s + (1.94 − 0.787i)15-s + (1.51 + 4.64i)16-s + (3.50 − 0.842i)17-s + ⋯ |
| L(s) = 1 | + (−0.586 − 1.15i)2-s + (0.226 + 0.974i)3-s + (−0.392 + 0.540i)4-s + (−0.0846 − 0.534i)5-s + (0.988 − 0.831i)6-s + (−0.287 + 0.245i)7-s + (−0.423 − 0.0670i)8-s + (−0.897 + 0.440i)9-s + (−0.565 + 0.410i)10-s + (0.667 + 0.409i)11-s + (−0.615 − 0.260i)12-s + (−0.0895 + 0.00704i)13-s + (0.451 + 0.186i)14-s + (0.501 − 0.203i)15-s + (0.377 + 1.16i)16-s + (0.851 − 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11782 - 0.192263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11782 - 0.192263i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.391 - 1.68i)T \) |
| 7 | \( 1 + (0.760 - 0.649i)T \) |
| 41 | \( 1 + (2.94 - 5.68i)T \) |
| good | 2 | \( 1 + (0.829 + 1.62i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (0.189 + 1.19i)T + (-4.75 + 1.54i)T^{2} \) |
| 11 | \( 1 + (-2.21 - 1.35i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (0.322 - 0.0254i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (-3.50 + 0.842i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-1.62 - 0.128i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (0.0319 - 0.0982i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.08 - 0.500i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (-4.63 - 6.37i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.99 - 2.17i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (-4.81 + 2.45i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-0.0426 + 0.0499i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (0.475 - 1.97i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-8.38 - 2.72i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (0.685 - 1.34i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-6.25 + 3.83i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (3.15 - 5.15i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (-6.19 - 6.19i)T + 73iT^{2} \) |
| 79 | \( 1 + (-6.94 + 2.87i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 6.50iT - 83T^{2} \) |
| 89 | \( 1 + (5.55 + 6.50i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (1.99 + 3.24i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−10.04132937014929672351690851211, −9.496718230309168641682989092223, −8.825421205427688275023470981136, −8.115177484587118685335122455915, −6.63957621923121533904679389158, −5.48776312953678758054178312032, −4.50907422269836624822989494998, −3.44824442258018495373743854785, −2.62050860462683252820057810285, −1.11211012572044258717237900785,
0.820582257181366168481730204609, 2.65089526999018838518006333931, 3.64675134110796104695385493664, 5.45868020285432093744218981265, 6.31054983130583932393742722056, 6.83697683460777433883657791055, 7.62654160422738745314601917683, 8.220173887101605039198626986049, 9.105447300985124214562424202285, 9.859460371882855137298383994787
