L(s) = 1 | + 2.06i·2-s + (−1.71 + 0.263i)3-s − 2.25·4-s + 5-s + (−0.543 − 3.53i)6-s − 0.527i·8-s + (2.86 − 0.902i)9-s + 2.06i·10-s + 4.69i·11-s + (3.86 − 0.594i)12-s − 0.638i·13-s + (−1.71 + 0.263i)15-s − 3.42·16-s − 4.14·17-s + (1.86 + 5.90i)18-s + 0.897i·19-s + ⋯ |
L(s) = 1 | + 1.45i·2-s + (−0.988 + 0.152i)3-s − 1.12·4-s + 0.447·5-s + (−0.221 − 1.44i)6-s − 0.186i·8-s + (0.953 − 0.300i)9-s + 0.652i·10-s + 1.41i·11-s + (1.11 − 0.171i)12-s − 0.177i·13-s + (−0.442 + 0.0680i)15-s − 0.855·16-s − 1.00·17-s + (0.438 + 1.39i)18-s + 0.205i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.311037 - 0.672466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.311037 - 0.672466i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 - 0.263i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.06iT - 2T^{2} \) |
| 11 | \( 1 - 4.69iT - 11T^{2} \) |
| 13 | \( 1 + 0.638iT - 13T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 - 0.897iT - 19T^{2} \) |
| 23 | \( 1 - 6.80iT - 23T^{2} \) |
| 29 | \( 1 + 2.14iT - 29T^{2} \) |
| 31 | \( 1 + 2.33iT - 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 4.10T + 41T^{2} \) |
| 43 | \( 1 - 3.14T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 + 2.26iT - 53T^{2} \) |
| 59 | \( 1 + 0.508T + 59T^{2} \) |
| 61 | \( 1 + 5.18iT - 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 - 1.22iT - 71T^{2} \) |
| 73 | \( 1 - 14.4iT - 73T^{2} \) |
| 79 | \( 1 - 9.08T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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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
−10.87233215288459144789796837511, −9.832321776022257321595126815676, −9.273111117718017302664179984083, −8.037881264319636965739849409914, −7.10459411732242341410690876334, −6.65307939072541459768837833694, −5.64437757797467515717200518110, −5.03744770484602062807287714870, −4.10891053950311244308239800722, −1.93667426773361517833534150686,
0.42470534176295551270422980209, 1.73050301217485834139540945815, 2.94218585532952878106329346721, 4.16146191847523603457379636998, 5.12147369320486848741913214905, 6.24206395983990372953512385583, 6.92269941608044261563688665632, 8.506827564029347615986851879192, 9.229166197604573244514374450323, 10.32960714007274752285686077724