L(s) = 1 | + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−1.86 − 0.897i)5-s + (2.21 − 1.45i)7-s + (−0.433 + 0.900i)8-s + (−0.897 − 1.86i)10-s + (−4.86 − 3.87i)11-s + (0.0474 + 0.0378i)13-s + (2.63 + 0.244i)14-s + (−0.900 + 0.433i)16-s + (1.28 − 5.61i)17-s − 2.01i·19-s + (0.460 − 2.01i)20-s + (−1.38 − 6.06i)22-s + (−0.713 + 0.162i)23-s + ⋯ |
L(s) = 1 | + (0.552 + 0.440i)2-s + (0.111 + 0.487i)4-s + (−0.833 − 0.401i)5-s + (0.836 − 0.548i)7-s + (−0.153 + 0.318i)8-s + (−0.283 − 0.589i)10-s + (−1.46 − 1.16i)11-s + (0.0131 + 0.0105i)13-s + (0.704 + 0.0654i)14-s + (−0.225 + 0.108i)16-s + (0.310 − 1.36i)17-s − 0.462i·19-s + (0.102 − 0.450i)20-s + (−0.295 − 1.29i)22-s + (−0.148 + 0.0339i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23714 - 0.804756i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23714 - 0.804756i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.21 + 1.45i)T \) |
good | 5 | \( 1 + (1.86 + 0.897i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (4.86 + 3.87i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.0474 - 0.0378i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.28 + 5.61i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 + 2.01iT - 19T^{2} \) |
| 23 | \( 1 + (0.713 - 0.162i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-8.50 - 1.94i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + 0.294iT - 31T^{2} \) |
| 37 | \( 1 + (0.570 - 2.49i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (7.61 + 3.66i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.03 - 0.500i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-3.82 + 4.79i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (7.87 - 1.79i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-4.79 + 2.30i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-4.45 - 1.01i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 4.72T + 67T^{2} \) |
| 71 | \( 1 + (1.80 - 0.412i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-12.0 + 9.60i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + 4.58T + 79T^{2} \) |
| 83 | \( 1 + (-0.413 - 0.518i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-6.55 - 8.21i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 14.0iT - 97T^{2} \) |
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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
−10.16235958867130593297381760786, −8.733133636807891782982201754284, −8.114091753180573174372076024042, −7.58073951153833561633931079649, −6.59771468682184742114793038788, −5.19140761780261637798427285614, −4.92142171389320631898205235285, −3.70944540580090428887100893035, −2.68326212466464719895868074309, −0.58581744733378142996186019809,
1.78909002098735962728218280573, 2.82502914032892148150823739661, 4.01944737301337215508374607197, 4.86392034667808960492565754334, 5.69515231175480476064251608929, 6.87503954975009162660418284946, 7.955045401605422795369110424804, 8.264355086141501641630741698175, 9.775436586645329878643515404986, 10.47193789352110275534540497930