L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.499i)6-s + (4.09 − 2.36i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (4.09 + 2.36i)11-s − 0.999·12-s + (3.59 − 0.232i)13-s − 4.73·14-s + (−0.5 + 0.866i)16-s + (2.59 + 4.5i)17-s + 0.999i·18-s + (1.09 − 0.633i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (0.353 − 0.204i)6-s + (1.54 − 0.894i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (1.23 + 0.713i)11-s − 0.288·12-s + (0.997 − 0.0643i)13-s − 1.26·14-s + (−0.125 + 0.216i)16-s + (0.630 + 1.09i)17-s + 0.235i·18-s + (0.251 − 0.145i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.666551731\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.666551731\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.59 + 0.232i)T \) |
good | 7 | \( 1 + (-4.09 + 2.36i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.59 - 4.5i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 + 0.633i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.09 - 1.90i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 37 | \( 1 + (2.59 + 1.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.401 - 0.232i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.09 + 5.36i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 1.26iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (12 - 6.92i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.40 + 4.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.29 - 5.36i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.09 + 4.09i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 + (-2.19 - 1.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.216387440867836744230347612292, −8.452336689182959634352141398632, −7.82749551473247006180283146478, −6.98476160155860988396362278137, −6.07829124670115205132284388890, −4.97833521983088396506763165975, −4.09759702845009636101798731704, −3.58857897256405516883179885675, −1.77113694928293355156939454176, −1.15232794409164270912100710235,
1.04451124054960848865348154810, 1.75958434377654984468084238532, 3.09279991862754643048718570422, 4.49908253404808572890183658174, 5.41200375168885157617239214757, 6.04428309238695196799159404662, 6.82814355362773752000642498554, 7.80490232414169668092008001045, 8.381091557364121570780533302822, 8.919562043892483510031686414453