| 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
−12.18246950662394011589009041387, −11.27750972617567070833885638746, −10.33945813520409587240486749557, −9.454668211763149299186865831278, −8.212088532112500777350991663252, −6.91457176224546601176701398475, −6.23812454450196610933129876963, −5.89247200856330330163756525484, −2.65853662508882116386603284476, −1.49156112515501653514307734246,
1.03521809451543005513239130942, 3.60099838950836285532686072380, 5.12702519596024244573492741787, 5.99455002662040562257961416404, 7.07127830631252606376882471839, 8.994789033631697449227986750460, 9.430665984707849562211727769611, 10.11811846829201809358063663058, 11.00804030625600498575399614712, 11.80469231865629645478834856309
