L(s) = 1 | − 2-s + (1.17 + 1.27i)3-s + 4-s + (−1.07 + 1.95i)5-s + (−1.17 − 1.27i)6-s − 8-s + (−0.259 + 2.98i)9-s + (1.07 − 1.95i)10-s + 6.01i·11-s + (1.17 + 1.27i)12-s + 0.864·13-s + (−3.76 + 0.914i)15-s + 16-s − 2.84i·17-s + (0.259 − 2.98i)18-s + 7.12i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.675 + 0.737i)3-s + 0.5·4-s + (−0.482 + 0.875i)5-s + (−0.477 − 0.521i)6-s − 0.353·8-s + (−0.0865 + 0.996i)9-s + (0.341 − 0.619i)10-s + 1.81i·11-s + (0.337 + 0.368i)12-s + 0.239·13-s + (−0.971 + 0.236i)15-s + 0.250·16-s − 0.690i·17-s + (0.0612 − 0.704i)18-s + 1.63i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.100978771\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.100978771\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.17 - 1.27i)T \) |
| 5 | \( 1 + (1.07 - 1.95i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6.01iT - 11T^{2} \) |
| 13 | \( 1 - 0.864T + 13T^{2} \) |
| 17 | \( 1 + 2.84iT - 17T^{2} \) |
| 19 | \( 1 - 7.12iT - 19T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 29 | \( 1 + 9.40iT - 29T^{2} \) |
| 31 | \( 1 - 2.68iT - 31T^{2} \) |
| 37 | \( 1 + 6.81iT - 37T^{2} \) |
| 41 | \( 1 - 5.32T + 41T^{2} \) |
| 43 | \( 1 - 2.68iT - 43T^{2} \) |
| 47 | \( 1 - 3.12iT - 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 7.09T + 59T^{2} \) |
| 61 | \( 1 - 9.33iT - 61T^{2} \) |
| 67 | \( 1 + 6.56iT - 67T^{2} \) |
| 71 | \( 1 - 4.46iT - 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 - 1.28iT - 83T^{2} \) |
| 89 | \( 1 + 3.08T + 89T^{2} \) |
| 97 | \( 1 - 1.90T + 97T^{2} \) |
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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
−9.658612887997822969232427819172, −9.448995564451372328799026272506, −8.101295982830723147110304807043, −7.71404031462592127728127253453, −6.98331495625899944490843927689, −5.90937931612070332833771657366, −4.60387483175348795145379882591, −3.85377694325974888749849235711, −2.78338256826195865094911668666, −1.91006867373459928363812471432,
0.52162290362040432893778074515, 1.43403832149458729566738971802, 2.90464674268220467217586232528, 3.62843283845936447615745957360, 5.01760552719573656393243975124, 6.10410180199232433082290779110, 6.85626671298525285509191828385, 7.82740963619674363547238756297, 8.436110842658307759760550859079, 8.892377935847161539903152779450