| L(s) = 1 | + (0.602 + 1.62i)3-s + (−2.89 − 1.05i)5-s + (0.0578 − 0.328i)7-s + (−2.27 + 1.95i)9-s + (−0.0918 + 0.0334i)11-s + (−0.709 + 0.595i)13-s + (−0.0324 − 5.33i)15-s + (−1.04 − 1.81i)17-s + (0.0352 − 0.0609i)19-s + (0.567 − 0.103i)21-s + (−1.09 − 6.20i)23-s + (3.42 + 2.87i)25-s + (−4.54 − 2.51i)27-s + (−5.42 − 4.55i)29-s + (−1.69 − 9.59i)31-s + ⋯ |
| L(s) = 1 | + (0.347 + 0.937i)3-s + (−1.29 − 0.470i)5-s + (0.0218 − 0.123i)7-s + (−0.758 + 0.652i)9-s + (−0.0276 + 0.0100i)11-s + (−0.196 + 0.165i)13-s + (−0.00837 − 1.37i)15-s + (−0.253 − 0.439i)17-s + (0.00807 − 0.0139i)19-s + (0.123 − 0.0226i)21-s + (−0.228 − 1.29i)23-s + (0.685 + 0.575i)25-s + (−0.875 − 0.484i)27-s + (−1.00 − 0.844i)29-s + (−0.303 − 1.72i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.216772 - 0.323126i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.216772 - 0.323126i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.602 - 1.62i)T \) |
| good | 5 | \( 1 + (2.89 + 1.05i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.0578 + 0.328i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.0918 - 0.0334i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.709 - 0.595i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.04 + 1.81i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0352 + 0.0609i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.09 + 6.20i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.42 + 4.55i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.69 + 9.59i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.24 - 3.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.632 + 0.530i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (8.73 - 3.17i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.949 + 5.38i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 + (13.0 + 4.73i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.712 - 4.03i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.50 + 7.97i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.92 - 5.05i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.07 - 3.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.81 - 4.03i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-11.4 - 9.58i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.08 + 1.87i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.94 - 1.07i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.760402418806497966317019416694, −9.127915208139487520060112542961, −8.102098000183000584816657108711, −7.77617195711614488268232197795, −6.43770563749719365602542730153, −5.14292520035632808708885091068, −4.33814440912461190390478970119, −3.73971630618110689602694029343, −2.45879789275602076401604681544, −0.17576640866010718348814246910,
1.65694073708221093325643958015, 3.11405173208319535321306236522, 3.76125063471306030529131908562, 5.19971260165494530823546499135, 6.33962135905207178156890165120, 7.28379574497430353627011295273, 7.68311868973648367305937832777, 8.563494340946757954032176198444, 9.355808901013741376914537416023, 10.69301175460318736936556619678
