L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 0.999·6-s + (−0.866 − 2.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1 + 1.73i)11-s + (−0.866 − 0.499i)12-s + 4i·13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (−4.33 + 2.5i)17-s + (0.866 − 0.499i)18-s + (−2 + 3.46i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s − 0.408·6-s + (−0.327 − 0.944i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.301 + 0.522i)11-s + (−0.249 − 0.144i)12-s + 1.10i·13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (−1.05 + 0.606i)17-s + (0.204 − 0.117i)18-s + (−0.458 + 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.541336464\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.541336464\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + (4.33 - 2.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.33 - 2.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-5.5 - 9.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (10.3 - 6i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + T + 71T^{2} \) |
| 73 | \( 1 + (1.73 - i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-5.5 + 9.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−10.42802241030279830643472581846, −9.349216646049414639850759894051, −8.546229775557506339830088937443, −7.29853002975653046513195184013, −6.71807671915113362617175386803, −6.08292712785656322092625826152, −4.68745019040997795796992304092, −4.32871316566124276510748097831, −3.26126471658803208745443960916, −1.59948932315091214143515757561,
0.60809531005130475931922103379, 2.36232564205064519064692102685, 3.10093320150734230222297993102, 4.56032970873753102977634176512, 5.25761743931894253841530279681, 6.27078368481508208528476928772, 6.68515767428171240815137792316, 8.028063827771741077000361669105, 8.879066153605457782529677333291, 9.736639216217508647996985829354