| L(s) = 1 | + (−1.41 − 0.00960i)2-s + (−1.76 + 2.30i)3-s + (1.99 + 0.0271i)4-s + (2.61 − 2.01i)5-s + (2.51 − 3.23i)6-s + (−1.37 − 2.25i)7-s + (−2.82 − 0.0576i)8-s + (−1.40 − 5.22i)9-s + (−3.72 + 2.81i)10-s + (3.36 − 0.443i)11-s + (−3.59 + 4.55i)12-s + (1.07 − 0.444i)13-s + (1.92 + 3.20i)14-s + 9.57i·15-s + (3.99 + 0.108i)16-s + (1.96 − 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.00679i)2-s + (−1.01 + 1.32i)3-s + (0.999 + 0.0135i)4-s + (1.17 − 0.898i)5-s + (1.02 − 1.32i)6-s + (−0.520 − 0.853i)7-s + (−0.999 − 0.0203i)8-s + (−0.466 − 1.74i)9-s + (−1.17 + 0.890i)10-s + (1.01 − 0.133i)11-s + (−1.03 + 1.31i)12-s + (0.297 − 0.123i)13-s + (0.514 + 0.857i)14-s + 2.47i·15-s + (0.999 + 0.0271i)16-s + (0.475 − 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.691896 - 0.00415849i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.691896 - 0.00415849i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.00960i)T \) |
| 7 | \( 1 + (1.37 + 2.25i)T \) |
| good | 3 | \( 1 + (1.76 - 2.30i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (-2.61 + 2.01i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-3.36 + 0.443i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 0.444i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-1.96 + 1.13i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.673 + 5.11i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (-2.29 - 8.54i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.58 - 3.81i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (0.413 + 0.716i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.692 + 0.531i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (-0.558 + 0.558i)T - 41iT^{2} \) |
| 43 | \( 1 + (-4.35 + 10.5i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (1.47 + 0.852i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.38 - 0.182i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (1.02 + 7.79i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.84 - 0.242i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (1.83 - 2.39i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-9.45 - 9.45i)T + 71iT^{2} \) |
| 73 | \( 1 + (6.77 + 1.81i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (10.6 + 6.14i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.62 + 2.74i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (3.05 - 0.819i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 6.07T + 97T^{2} \) |
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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
−11.80469231865629645478834856309, −11.00804030625600498575399614712, −10.11811846829201809358063663058, −9.430665984707849562211727769611, −8.994789033631697449227986750460, −7.07127830631252606376882471839, −5.99455002662040562257961416404, −5.12702519596024244573492741787, −3.60099838950836285532686072380, −1.03521809451543005513239130942,
1.49156112515501653514307734246, 2.65853662508882116386603284476, 5.89247200856330330163756525484, 6.23812454450196610933129876963, 6.91457176224546601176701398475, 8.212088532112500777350991663252, 9.454668211763149299186865831278, 10.33945813520409587240486749557, 11.27750972617567070833885638746, 12.18246950662394011589009041387
