L(s) = 1 | + (0.279 + 1.38i)2-s + (−1.84 + 0.774i)4-s + (−2.25 − 0.821i)5-s + (1.95 + 0.344i)7-s + (−1.58 − 2.34i)8-s + (0.508 − 3.35i)10-s + (−0.553 − 1.52i)11-s + (−3.57 − 4.26i)13-s + (0.0681 + 2.80i)14-s + (2.80 − 2.85i)16-s + (6.40 − 3.69i)17-s + (−1.39 + 2.41i)19-s + (4.79 − 0.233i)20-s + (1.95 − 1.19i)22-s + (−0.465 − 2.64i)23-s + ⋯ |
L(s) = 1 | + (0.197 + 0.980i)2-s + (−0.921 + 0.387i)4-s + (−1.00 − 0.367i)5-s + (0.738 + 0.130i)7-s + (−0.561 − 0.827i)8-s + (0.160 − 1.06i)10-s + (−0.166 − 0.458i)11-s + (−0.992 − 1.18i)13-s + (0.0182 + 0.749i)14-s + (0.700 − 0.714i)16-s + (1.55 − 0.897i)17-s + (−0.320 + 0.554i)19-s + (1.07 − 0.0521i)20-s + (0.416 − 0.254i)22-s + (−0.0970 − 0.550i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.933347 - 0.232097i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.933347 - 0.232097i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.279 - 1.38i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.25 + 0.821i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.95 - 0.344i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.553 + 1.52i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.57 + 4.26i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-6.40 + 3.69i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.465 + 2.64i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.138 + 0.116i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.76 - 0.311i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.55 + 2.05i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.85 - 3.40i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-11.1 + 4.04i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.08 + 11.8i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + (-1.96 + 5.40i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (7.60 + 1.34i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.38 + 1.16i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.25 + 2.16i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.79 - 6.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.77 - 8.06i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.718 - 0.855i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (9.87 + 5.70i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.25 - 1.91i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.36478688766611100609928179858, −9.440742013994822741716884388621, −8.244707496634187548721253165831, −7.923668613522279696684055415180, −7.24195817750993797263551818444, −5.76083505497032614700778180384, −5.13147999412350456932836130008, −4.16526436372941597098175179011, −3.02963775611055396536314470104, −0.52187707350079529598312162013,
1.54982392488305691111267717627, 2.88401679536777123415913706418, 4.10137186795255877633992144922, 4.65950976061829456391145656096, 5.91188047973405728872789513581, 7.45272722999729335973775601834, 7.88850367284360590117999680847, 9.133206234818501638066996085639, 9.875028559505879528628910075703, 10.90307173100661432214343498256