L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.58 + 2.75i)3-s + (−0.499 − 0.866i)4-s + (−1.58 − 2.75i)6-s − 4.17·7-s + 0.999·8-s + (−3.54 − 6.14i)9-s − 3.90·11-s + 3.17·12-s + (−0.239 − 0.414i)13-s + (2.08 − 3.61i)14-s + (−0.5 + 0.866i)16-s + (0.245 − 0.424i)17-s + 7.09·18-s + (2.74 + 3.38i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.916 + 1.58i)3-s + (−0.249 − 0.433i)4-s + (−0.648 − 1.12i)6-s − 1.57·7-s + 0.353·8-s + (−1.18 − 2.04i)9-s − 1.17·11-s + 0.916·12-s + (−0.0664 − 0.115i)13-s + (0.558 − 0.966i)14-s + (−0.125 + 0.216i)16-s + (0.0594 − 0.102i)17-s + 1.67·18-s + (0.628 + 0.777i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.270962 + 0.0506002i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.270962 + 0.0506002i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.74 - 3.38i)T \) |
good | 3 | \( 1 + (1.58 - 2.75i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 11 | \( 1 + 3.90T + 11T^{2} \) |
| 13 | \( 1 + (0.239 + 0.414i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.245 + 0.424i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.84 - 4.93i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.21 + 2.10i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.42T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + (-0.254 + 0.441i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.91 + 5.04i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.485 - 0.841i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.77 + 4.81i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.82 + 6.62i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.65 - 8.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.809 + 1.40i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.937 + 1.62i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.35 + 4.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.19 - 5.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.96T + 83T^{2} \) |
| 89 | \( 1 + (-2.76 - 4.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.24 + 12.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.01333962600850893000700863345, −9.547599998173975386232420225852, −8.703623429545677905720031819504, −7.44972347661551928760344577630, −6.46856161780979382460769201904, −5.59215199506106983485731195687, −5.18501216753764824530448976397, −3.88043378282599993466355818549, −3.09597177905790571359326024090, −0.23338922891995296613710341312,
0.814864309859528319226307051989, 2.36074436197290490234722644262, 3.12078680930285156551149169133, 4.91910170519139428814643336256, 5.86070199551607035816769249159, 6.76794770891426138608554856496, 7.26560816277112335412248125685, 8.226336269877036901088121359659, 9.189123522132042917717280330474, 10.25616478065153796372983276088