| L(s) = 1 | + (−1.09 + 0.899i)2-s + (1.16 + 2.27i)3-s + (0.380 − 1.96i)4-s + (0.211 + 2.22i)5-s + (−3.31 − 1.44i)6-s + 0.405·7-s + (1.35 + 2.48i)8-s + (−2.08 + 2.86i)9-s + (−2.23 − 2.23i)10-s + (−0.148 + 0.936i)11-s + (4.91 − 1.41i)12-s + (−1.95 + 0.310i)13-s + (−0.441 + 0.364i)14-s + (−4.82 + 3.06i)15-s + (−3.71 − 1.49i)16-s + (3.96 − 1.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.771 + 0.636i)2-s + (0.670 + 1.31i)3-s + (0.190 − 0.981i)4-s + (0.0944 + 0.995i)5-s + (−1.35 − 0.588i)6-s + 0.153·7-s + (0.477 + 0.878i)8-s + (−0.694 + 0.955i)9-s + (−0.706 − 0.707i)10-s + (−0.0447 + 0.282i)11-s + (1.41 − 0.407i)12-s + (−0.542 + 0.0859i)13-s + (−0.118 + 0.0974i)14-s + (−1.24 + 0.791i)15-s + (−0.927 − 0.373i)16-s + (0.960 − 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.170828 + 1.11584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170828 + 1.11584i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.09 - 0.899i)T \) |
| 5 | \( 1 + (-0.211 - 2.22i)T \) |
| good | 3 | \( 1 + (-1.16 - 2.27i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 - 0.405T + 7T^{2} \) |
| 11 | \( 1 + (0.148 - 0.936i)T + (-10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (1.95 - 0.310i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-3.96 + 1.28i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.51 - 4.94i)T + (-11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (-4.30 + 3.12i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (4.08 - 2.07i)T + (17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (0.642 + 1.97i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.08 + 6.83i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-6.12 + 8.43i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.87 - 4.87i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.222 - 0.0721i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (6.20 - 3.16i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (3.94 - 0.625i)T + (56.1 - 18.2i)T^{2} \) |
| 61 | \( 1 + (4.88 + 0.774i)T + (58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (-9.11 - 4.64i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-8.84 - 2.87i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.31 + 1.68i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.24 + 9.99i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-15.9 - 8.14i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-8.28 - 11.4i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.21 - 1.37i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.98175941282210181257629716311, −10.58750347520742227660437032440, −9.683384816141792996277677501964, −9.217360828139405707017097644590, −8.008722471192929524459651402923, −7.28032212794169362278176006396, −6.03791378060968626692037842185, −4.93842657707870365118381711679, −3.67078636681532312477587306847, −2.33648103945513433017521431190,
0.917988842522864316692316867905, 2.02649971023550003927509778924, 3.24861821607051865656913411589, 4.87278550121692776330679413513, 6.46543681615402966181690941284, 7.59630333600045762169230161590, 8.071525882494226688344981270648, 8.978285994868084656149663193583, 9.641938057142272926588993677040, 10.98615235157553000389019599170
