L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.499i)6-s + (1.5 − 2.59i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (0.232 − 0.133i)11-s − 0.999i·12-s + (3.5 + 0.866i)13-s − 3·14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s − 0.999·18-s + (−4.96 − 2.86i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.353 + 0.204i)6-s + (0.566 − 0.981i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.0699 − 0.0403i)11-s − 0.288i·12-s + (0.970 + 0.240i)13-s − 0.801·14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s − 0.235·18-s + (−1.13 − 0.657i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.270829726\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270829726\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.5 - 0.866i)T \) |
good | 7 | \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.232 + 0.133i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.96 + 2.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 1.73i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.732 - 1.26i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.92iT - 31T^{2} \) |
| 37 | \( 1 + (-2.96 - 5.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.46 - 2i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.19 + 3i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.46T + 47T^{2} \) |
| 53 | \( 1 + 0.267iT - 53T^{2} \) |
| 59 | \( 1 + (-9.92 - 5.73i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.267 + 0.464i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.732 + 1.26i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.1 + 6.46i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 3.07T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 + (-12.2 + 7.06i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.19 + 7.26i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.978986090467798706395430859642, −8.433641941386513725511761573867, −7.53205375876551158688981311365, −6.69671762119779998915690477180, −5.82089796822931547569123152291, −4.64255641108054443677641647082, −4.12490396768012320428632664951, −3.15990392642374962054066467665, −1.70999028271412598695349522804, −0.70118150265477416887726142717,
1.08076511406329344796919726155, 2.19344963941081354114626516443, 3.60301788456713829409251235302, 4.82176699759887838096354829144, 5.55901273552213636163081027197, 6.10982331571858888452229545531, 6.97680977569169646796853536017, 7.85928740090406698163979071008, 8.519698527619795107517255784089, 9.079671160023642177099906488504