| L(s) = 1 | + (−1.30 − 0.546i)2-s + (−0.540 + 0.841i)3-s + (1.40 + 1.42i)4-s + (−0.190 − 0.0867i)5-s + (1.16 − 0.801i)6-s + (2.41 + 0.710i)7-s + (−1.04 − 2.62i)8-s + (−0.415 − 0.909i)9-s + (0.200 + 0.217i)10-s + (−4.36 − 3.78i)11-s + (−1.95 + 0.408i)12-s + (−1.69 − 5.78i)13-s + (−2.76 − 2.24i)14-s + (0.175 − 0.112i)15-s + (−0.0697 + 3.99i)16-s + (0.569 + 3.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.922 − 0.386i)2-s + (−0.312 + 0.485i)3-s + (0.700 + 0.713i)4-s + (−0.0849 − 0.0388i)5-s + (0.475 − 0.327i)6-s + (0.914 + 0.268i)7-s + (−0.370 − 0.928i)8-s + (−0.138 − 0.303i)9-s + (0.0633 + 0.0686i)10-s + (−1.31 − 1.14i)11-s + (−0.565 + 0.117i)12-s + (−0.471 − 1.60i)13-s + (−0.739 − 0.600i)14-s + (0.0453 − 0.0291i)15-s + (−0.0174 + 0.999i)16-s + (0.138 + 0.960i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.643726 - 0.412974i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643726 - 0.412974i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 + 0.546i)T \) |
| 3 | \( 1 + (0.540 - 0.841i)T \) |
| 23 | \( 1 + (-3.93 + 2.74i)T \) |
| good | 5 | \( 1 + (0.190 + 0.0867i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-2.41 - 0.710i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (4.36 + 3.78i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.69 + 5.78i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.569 - 3.95i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-6.83 - 0.982i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.673 - 0.0969i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.246 + 0.158i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.94 - 0.889i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.71 + 10.3i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-3.10 + 4.83i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 3.14T + 47T^{2} \) |
| 53 | \( 1 + (0.724 - 2.46i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (2.17 + 7.41i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-6.75 - 10.5i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-2.83 + 2.45i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-7.73 - 8.92i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.62 + 11.3i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (2.27 - 0.669i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-13.0 + 5.94i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (14.0 + 9.05i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.56 - 5.60i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.60202927865905802042960277000, −10.01726592016342313913321362304, −8.763509738058667772136987586378, −8.090419209453091254155215174482, −7.49803840633279330514385157291, −5.82273037730674110085992939581, −5.21127196131488331765988954681, −3.56449031507653000230689537878, −2.55419662057193482545487707396, −0.66454871265140197538370764995,
1.39296664963390161936111070109, 2.54497766363176947450800999537, 4.79664771401000934647273269521, 5.35432022570050800416883422740, 6.88145145050526578548531625835, 7.43984125795466108979838044337, 7.932201343206749130505466758450, 9.406618220577228840241037009303, 9.744229344326741870112567372224, 11.18928837282953267589482656749
