| L(s) = 1 | + (−0.688 + 1.89i)2-s + (−1.40 + 1.01i)3-s + (−1.57 − 1.31i)4-s + (−0.535 − 2.17i)5-s + (−0.953 − 3.35i)6-s + (−1.02 − 0.590i)7-s + (0.0898 − 0.0518i)8-s + (0.939 − 2.84i)9-s + (4.47 + 0.481i)10-s + (0.811 − 0.468i)11-s + (3.54 + 0.255i)12-s + (0.827 − 4.69i)13-s + (1.81 − 1.52i)14-s + (2.95 + 2.50i)15-s + (−0.676 − 3.83i)16-s + (2.40 + 0.875i)17-s + ⋯ |
| L(s) = 1 | + (−0.486 + 1.33i)2-s + (−0.810 + 0.585i)3-s + (−0.785 − 0.659i)4-s + (−0.239 − 0.970i)5-s + (−0.389 − 1.36i)6-s + (−0.386 − 0.223i)7-s + (0.0317 − 0.0183i)8-s + (0.313 − 0.949i)9-s + (1.41 + 0.152i)10-s + (0.244 − 0.141i)11-s + (1.02 + 0.0738i)12-s + (0.229 − 1.30i)13-s + (0.486 − 0.408i)14-s + (0.762 + 0.646i)15-s + (−0.169 − 0.958i)16-s + (0.583 + 0.212i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.428701 - 0.0550064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428701 - 0.0550064i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.40 - 1.01i)T \) |
| 5 | \( 1 + (0.535 + 2.17i)T \) |
| 19 | \( 1 + (3.67 - 2.34i)T \) |
| good | 2 | \( 1 + (0.688 - 1.89i)T + (-1.53 - 1.28i)T^{2} \) |
| 7 | \( 1 + (1.02 + 0.590i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.811 + 0.468i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.827 + 4.69i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.40 - 0.875i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (4.80 + 4.03i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.79 + 2.47i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.55 - 0.897i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.77T + 37T^{2} \) |
| 41 | \( 1 + (1.08 + 6.13i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.40 + 8.82i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.22 + 0.446i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.51 + 2.99i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (6.90 + 2.51i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.529 - 0.444i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.25 + 2.27i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (5.19 - 4.35i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.891 + 0.157i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-9.32 + 1.64i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.08 + 1.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.68 - 15.2i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (8.87 + 3.22i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.01130723486634015351735252657, −10.49001627436816588990810076093, −9.844154306207021257893778665438, −8.610431767363646799741686113847, −8.096245962161014606109162976656, −6.71003442410831753599061899000, −5.87476424656350173592598559797, −5.06348761591455378245225043595, −3.76610992489582231623657901223, −0.42429220367816273396263793631,
1.67218781450366157057697813624, 2.93588295420805560896549431951, 4.33660072307094015343267730458, 6.20601358613364407834978443760, 6.78483576195908722902284759574, 8.125549496601138821983311346758, 9.451867161956573446126283808003, 10.24013736616002635862565943216, 11.09360454574740262295605914387, 11.74386231770087744031889299223
