L(s) = 1 | + (−0.533 + 1.30i)2-s + (−0.910 − 0.377i)3-s + (−1.43 − 1.39i)4-s + (−0.227 − 0.549i)5-s + (0.979 − 0.991i)6-s + (−0.707 + 0.707i)7-s + (2.59 − 1.12i)8-s + (−1.43 − 1.43i)9-s + (0.840 − 0.00493i)10-s + (4.11 − 1.70i)11-s + (0.775 + 1.81i)12-s + (2.17 − 5.25i)13-s + (−0.548 − 1.30i)14-s + 0.586i·15-s + (0.0938 + 3.99i)16-s − 4.07i·17-s + ⋯ |
L(s) = 1 | + (−0.377 + 0.926i)2-s + (−0.525 − 0.217i)3-s + (−0.715 − 0.698i)4-s + (−0.101 − 0.245i)5-s + (0.400 − 0.404i)6-s + (−0.267 + 0.267i)7-s + (0.917 − 0.398i)8-s + (−0.478 − 0.478i)9-s + (0.265 − 0.00155i)10-s + (1.23 − 0.513i)11-s + (0.223 + 0.523i)12-s + (0.604 − 1.45i)13-s + (−0.146 − 0.348i)14-s + 0.151i·15-s + (0.0234 + 0.999i)16-s − 0.988i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.668742 - 0.192200i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.668742 - 0.192200i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.533 - 1.30i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.910 + 0.377i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.227 + 0.549i)T + (-3.53 + 3.53i)T^{2} \) |
| 11 | \( 1 + (-4.11 + 1.70i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-2.17 + 5.25i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 4.07iT - 17T^{2} \) |
| 19 | \( 1 + (0.900 - 2.17i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (0.759 + 0.759i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.59 + 1.07i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 5.93T + 31T^{2} \) |
| 37 | \( 1 + (-4.05 - 9.78i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (7.57 + 7.57i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.08 + 0.864i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 5.52iT - 47T^{2} \) |
| 53 | \( 1 + (-4.99 + 2.06i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (1.25 + 3.02i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (4.79 + 1.98i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-7.30 - 3.02i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (7.81 - 7.81i)T - 71iT^{2} \) |
| 73 | \( 1 + (-8.65 - 8.65i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.2iT - 79T^{2} \) |
| 83 | \( 1 + (-2.50 + 6.05i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.86 + 6.86i)T - 89iT^{2} \) |
| 97 | \( 1 - 0.269T + 97T^{2} \) |
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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
−12.18134311649484038554367421995, −11.20493533406501362355534480765, −10.05731546651358146223759683356, −8.953783120877709576534170625665, −8.314149959845929106907499957524, −6.92687436360356391715442625473, −6.07494070775526024937524121597, −5.27880900329925395974145137588, −3.59681257081688624689665504257, −0.75199280668690443628240543398,
1.79225754082998243082099733595, 3.64641923832735779655998871580, 4.58004817843794707191034695272, 6.22401232003830314555500003904, 7.38656088937871283353779072027, 8.824730323063624831287252964291, 9.435065228052067457946218885739, 10.73943543878058751355227349330, 11.20839794472346467798551264490, 12.03512339357987881927516651578