| L(s) = 1 | + (−0.978 − 0.207i)2-s + (−1.12 − 1.94i)3-s + (0.913 + 0.406i)4-s + (0.0802 + 0.763i)5-s + (0.693 + 2.13i)6-s + (2.61 − 0.430i)7-s + (−0.809 − 0.587i)8-s + (−1.01 + 1.76i)9-s + (0.0802 − 0.763i)10-s + (−0.600 + 5.71i)11-s + (−0.234 − 2.23i)12-s + (0.0730 + 0.224i)13-s + (−2.64 − 0.121i)14-s + (1.39 − 1.01i)15-s + (0.669 + 0.743i)16-s + (−0.629 + 5.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.647 − 1.12i)3-s + (0.456 + 0.203i)4-s + (0.0358 + 0.341i)5-s + (0.283 + 0.871i)6-s + (0.986 − 0.162i)7-s + (−0.286 − 0.207i)8-s + (−0.339 + 0.588i)9-s + (0.0253 − 0.241i)10-s + (−0.180 + 1.72i)11-s + (−0.0677 − 0.644i)12-s + (0.0202 + 0.0623i)13-s + (−0.706 − 0.0324i)14-s + (0.359 − 0.261i)15-s + (0.167 + 0.185i)16-s + (−0.152 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.874689 + 0.0492032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874689 + 0.0492032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (-2.61 + 0.430i)T \) |
| 41 | \( 1 + (-5.53 - 3.22i)T \) |
| good | 3 | \( 1 + (1.12 + 1.94i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.0802 - 0.763i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (0.600 - 5.71i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.0730 - 0.224i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.629 - 5.99i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-2.80 - 3.11i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (4.27 + 0.907i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-2.22 + 1.61i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.14 - 10.8i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.664 + 6.32i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (2.21 + 6.81i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-6.05 - 1.28i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-6.53 - 2.90i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-6.73 + 7.48i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.87 - 2.07i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (4.10 + 1.82i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (5.86 + 4.26i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.334 + 0.579i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.923 + 1.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + (-2.92 - 3.24i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-3.33 + 2.42i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.58777787267037805367528543919, −10.21555485359581085795384972957, −8.839240161634929730184475571311, −7.85573208430087090893603824753, −7.27091004950400827936854805073, −6.52557202251075742602282860479, −5.42326963684980557419692594845, −4.09934289668187780366959393797, −2.15876896884862860603991408401, −1.38780021059406019412951753157,
0.75397140443765414688209469444, 2.78524685045909506576005141457, 4.30047931826122182445682264733, 5.28847195007102800339380208593, 5.83014710286722376861864357672, 7.29791815447614440351025262258, 8.295193321272389559295096392500, 9.024636382136847420204016071492, 9.820714913559086622446713321543, 10.75104434655179384061100958842
