L(s) = 1 | + (1.38 + 2.39i)5-s + 3.26i·7-s + 1.94i·11-s + (3.96 + 2.29i)13-s + (−2.27 − 3.94i)17-s + (−3.43 + 2.68i)19-s + (7.81 + 4.51i)23-s + (−1.32 + 2.29i)25-s + (1.78 + 1.03i)29-s − 1.21·31-s + (−7.81 + 4.51i)35-s − 1.73i·37-s + (3.25 − 1.87i)41-s + (−1.82 + 1.05i)43-s + (9.49 + 5.48i)47-s + ⋯ |
L(s) = 1 | + (0.618 + 1.07i)5-s + 1.23i·7-s + 0.585i·11-s + (1.10 + 0.635i)13-s + (−0.551 − 0.955i)17-s + (−0.787 + 0.615i)19-s + (1.62 + 0.940i)23-s + (−0.264 + 0.458i)25-s + (0.331 + 0.191i)29-s − 0.218·31-s + (−1.32 + 0.762i)35-s − 0.284i·37-s + (0.508 − 0.293i)41-s + (−0.278 + 0.160i)43-s + (1.38 + 0.799i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.010804966\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.010804966\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.43 - 2.68i)T \) |
good | 5 | \( 1 + (-1.38 - 2.39i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 3.26iT - 7T^{2} \) |
| 11 | \( 1 - 1.94iT - 11T^{2} \) |
| 13 | \( 1 + (-3.96 - 2.29i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.27 + 3.94i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-7.81 - 4.51i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.78 - 1.03i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 + 1.73iT - 37T^{2} \) |
| 41 | \( 1 + (-3.25 + 1.87i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.82 - 1.05i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-9.49 - 5.48i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.40 + 4.27i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.68 + 2.91i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.322 + 0.559i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.04 - 7.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.13 + 10.6i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.79 + 6.56i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.04 - 8.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.9iT - 83T^{2} \) |
| 89 | \( 1 + (-5.93 - 3.42i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (16.4 - 9.47i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.203315932772426769755774898737, −8.503111732668643269518559896640, −7.39022179968370684955022236782, −6.70992772809835744540297474152, −6.10088502515042720964154001059, −5.37478890446128233705626045379, −4.38029955771327948454991367617, −3.20433234596482913355076381118, −2.49026912430652368378612046536, −1.62732348809358775981590381322,
0.68550813497058045684816252955, 1.39880528352425649442222884101, 2.81527864928972068746039496259, 3.94445280785223250896696933546, 4.55416833220302720786337682861, 5.46665908849307690174796230025, 6.27103502103210773723175280985, 6.93682846599514127686504833697, 7.997821092305572905433651666660, 8.728535772919580765496248338796