L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 + 1.5i)3-s + (−0.499 − 0.866i)4-s − 1.73·6-s + (−0.866 + 2.5i)7-s + 0.999·8-s + (−1.5 + 2.59i)9-s + (−5.19 + 3i)11-s + (0.866 − 1.49i)12-s + 1.73·13-s + (−1.73 − 2i)14-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)18-s + (1.5 + 0.866i)19-s + (−4.5 + 0.866i)21-s − 6i·22-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.499 + 0.866i)3-s + (−0.249 − 0.433i)4-s − 0.707·6-s + (−0.327 + 0.944i)7-s + 0.353·8-s + (−0.5 + 0.866i)9-s + (−1.56 + 0.904i)11-s + (0.250 − 0.433i)12-s + 0.480·13-s + (−0.462 − 0.534i)14-s + (−0.125 + 0.216i)16-s + (−0.353 − 0.612i)18-s + (0.344 + 0.198i)19-s + (−0.981 + 0.188i)21-s − 1.27i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8061192711\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8061192711\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.866 - 2.5i)T \) |
good | 11 | \( 1 + (5.19 - 3i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.06 - 3.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (9 + 5.19i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.19 - 9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.52 - 5.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (0.866 + 1.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−10.04566281438624587054093217375, −9.673709992963117261600586361820, −8.596667778121878209380333404275, −8.213694711531405741526933172993, −7.23197866819378619310071388504, −6.07751855666193197028116288219, −5.21672406563922790306268294420, −4.56457882886117855552557233344, −3.13401263484360007462637180421, −2.21390849959634731343790203412,
0.37019348440403737713809300688, 1.62360147496891683744330149361, 3.05597221305290664152663444568, 3.47586365344604612828805817798, 5.04179991438019777395731392910, 6.13561580744863693479789277426, 7.25921498553290380656404625015, 7.74576778124761093726619774894, 8.571079299617926779538705797863, 9.362855084097095050129941106592