L(s) = 1 | + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (−3.14 − 2.14i)5-s + (−0.923 − 2.47i)7-s + (−0.433 − 0.900i)8-s + (2.14 + 3.14i)10-s + (−0.0766 − 0.508i)11-s + (1.07 − 0.856i)13-s + (−0.0461 + 2.64i)14-s + (0.0747 + 0.997i)16-s + (−4.86 − 1.50i)17-s + (0.729 − 0.421i)19-s + (−0.847 − 3.71i)20-s + (−0.114 + 0.501i)22-s + (0.673 + 2.18i)23-s + ⋯ |
L(s) = 1 | + (−0.658 − 0.258i)2-s + (0.366 + 0.340i)4-s + (−1.40 − 0.959i)5-s + (−0.349 − 0.937i)7-s + (−0.153 − 0.318i)8-s + (0.678 + 0.995i)10-s + (−0.0231 − 0.153i)11-s + (0.297 − 0.237i)13-s + (−0.0123 + 0.706i)14-s + (0.0186 + 0.249i)16-s + (−1.18 − 0.364i)17-s + (0.167 − 0.0966i)19-s + (−0.189 − 0.830i)20-s + (−0.0244 + 0.106i)22-s + (0.140 + 0.455i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0278429 + 0.0424437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0278429 + 0.0424437i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.930 + 0.365i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.923 + 2.47i)T \) |
good | 5 | \( 1 + (3.14 + 2.14i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.0766 + 0.508i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 0.856i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (4.86 + 1.50i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.729 + 0.421i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.673 - 2.18i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-1.32 + 0.302i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.29 + 1.32i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.31 - 6.78i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-9.45 + 4.55i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-5.07 - 2.44i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (4.42 - 11.2i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (7.50 - 8.08i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-5.17 + 3.53i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (3.45 + 3.72i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (1.69 - 2.93i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.06 + 2.06i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (2.42 - 0.953i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (6.41 + 11.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.24 + 9.08i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (6.05 + 0.912i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 1.21iT - 97T^{2} \) |
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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
−9.333048219036153989560283210953, −8.820278552372014942926925931763, −7.81044210479978541764969138404, −7.41673275300303134922015370045, −6.31317207848167953399904414984, −4.78761802583063759644922241609, −4.06143866727742394062651526711, −3.09453694297426423087879854857, −1.18502177978233664689922728032, −0.03394465699285857798240141476,
2.22632445518788575820850555926, 3.30748437178631125149195745329, 4.36450772461237060132052988062, 5.75400898353912054713147625609, 6.74522792327049533696470471315, 7.22839378104248218208729012958, 8.334719796941781239530891636055, 8.781971083811706031079912634657, 9.831492561996032082359937441005, 10.87355525958830467534855816362