L(s) = 1 | + (−0.5 − 0.866i)2-s + (−1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 1.73i·6-s + (0.866 + 2.5i)7-s + 0.999·8-s + (1.5 + 2.59i)9-s + (2.59 + 1.5i)11-s + (1.49 − 0.866i)12-s − 3.46·13-s + (1.73 − 2i)14-s + (−0.5 − 0.866i)16-s + (−3 − 1.73i)17-s + (1.5 − 2.59i)18-s + (−3 + 1.73i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.707i·6-s + (0.327 + 0.944i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (0.783 + 0.452i)11-s + (0.433 − 0.250i)12-s − 0.960·13-s + (0.462 − 0.534i)14-s + (−0.125 − 0.216i)16-s + (−0.727 − 0.420i)17-s + (0.353 − 0.612i)18-s + (−0.688 + 0.397i)19-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 2.5i)T \) |
good | 11 | \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + (3 + 1.73i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + (6 - 3.46i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + (3.46 - 6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.66iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−9.527525782557878762833750420275, −8.660342234610099828162825295810, −7.81416323922721533256631797812, −6.85119087835129809259418569431, −6.07794352423813635703091143968, −4.96170068342187750989397367121, −4.24818510994907391555937554864, −2.49790404754263536251347492562, −1.76511693994370168460518702156, 0,
1.51577908865055299190380275351, 3.62745866484663030892298841633, 4.50345347372588764675566966873, 5.24494937344771526425172908541, 6.39551878191952648370641495400, 6.85760198257489910986611233812, 7.83284243464519478021092109416, 8.816995634512398153659368743522, 9.696435920478645446863270124350