| L(s) = 1 | + (−0.800 − 0.599i)2-s + (−2.81 + 1.04i)3-s + (0.281 + 0.959i)4-s + (0.282 + 2.21i)5-s + (2.88 + 0.845i)6-s + (−1.98 − 0.431i)7-s + (0.349 − 0.936i)8-s + (4.54 − 3.93i)9-s + (1.10 − 1.94i)10-s + (−0.731 − 0.105i)11-s + (−1.79 − 2.40i)12-s + (−1.02 − 4.71i)13-s + (1.32 + 1.53i)14-s + (−3.12 − 5.94i)15-s + (−0.841 + 0.540i)16-s + (−0.560 − 1.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.566 − 0.423i)2-s + (−1.62 + 0.605i)3-s + (0.140 + 0.479i)4-s + (0.126 + 0.991i)5-s + (1.17 + 0.345i)6-s + (−0.749 − 0.163i)7-s + (0.123 − 0.331i)8-s + (1.51 − 1.31i)9-s + (0.348 − 0.615i)10-s + (−0.220 − 0.0316i)11-s + (−0.519 − 0.693i)12-s + (−0.284 − 1.30i)13-s + (0.355 + 0.409i)14-s + (−0.805 − 1.53i)15-s + (−0.210 + 0.135i)16-s + (−0.135 − 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0640837 - 0.126729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0640837 - 0.126729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.800 + 0.599i)T \) |
| 5 | \( 1 + (-0.282 - 2.21i)T \) |
| 23 | \( 1 + (-3.16 + 3.60i)T \) |
| good | 3 | \( 1 + (2.81 - 1.04i)T + (2.26 - 1.96i)T^{2} \) |
| 7 | \( 1 + (1.98 + 0.431i)T + (6.36 + 2.90i)T^{2} \) |
| 11 | \( 1 + (0.731 + 0.105i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.02 + 4.71i)T + (-11.8 + 5.40i)T^{2} \) |
| 17 | \( 1 + (0.560 + 1.02i)T + (-9.19 + 14.3i)T^{2} \) |
| 19 | \( 1 + (4.62 - 1.35i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.24 + 7.66i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.40 + 5.26i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.671 - 9.38i)T + (-36.6 - 5.26i)T^{2} \) |
| 41 | \( 1 + (-4.64 + 5.36i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.250 - 0.670i)T + (-32.4 + 28.1i)T^{2} \) |
| 47 | \( 1 + (2.04 - 2.04i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.00 - 4.61i)T + (-48.2 - 22.0i)T^{2} \) |
| 59 | \( 1 + (5.31 - 8.27i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (10.6 - 4.87i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-4.97 + 6.64i)T + (-18.8 - 64.2i)T^{2} \) |
| 71 | \( 1 + (1.65 + 11.4i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (10.7 + 5.89i)T + (39.4 + 61.4i)T^{2} \) |
| 79 | \( 1 + (2.40 + 1.54i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (7.88 + 0.564i)T + (82.1 + 11.8i)T^{2} \) |
| 89 | \( 1 + (4.10 - 8.98i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.85 + 0.204i)T + (96.0 - 13.8i)T^{2} \) |
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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
−11.59828704520660520912863638420, −10.52613641299905059956563983287, −10.46347587052649324005678957129, −9.470977308001680177009133622242, −7.76157800868782981758525329607, −6.55857825101466656669644451326, −5.88662084885384621850603531958, −4.38024750065302088860877462326, −2.92073503773287234622078836155, −0.16990004659376939617852954153,
1.58217496598771703932239639856, 4.60129574505813413149690587957, 5.54625400750258555476253599974, 6.51317074001947380542165836618, 7.21117045474003473539541759385, 8.706502025069370219917721004739, 9.574066969161770328064696069654, 10.74608660094502120688050944804, 11.54445325518426033970578244201, 12.65836426611443308336516313444
