| L(s) = 1 | + (−0.0744 − 1.41i)2-s + (0.429 − 0.165i)3-s + (−1.98 + 0.210i)4-s + (−1.13 + 1.45i)5-s + (−0.266 − 0.594i)6-s + (1.18 − 3.30i)7-s + (0.444 + 2.79i)8-s + (−2.06 + 1.87i)9-s + (2.13 + 1.49i)10-s + (−0.943 − 4.20i)11-s + (−0.820 + 0.420i)12-s + (−6.00 + 1.66i)13-s + (−4.75 − 1.42i)14-s + (−0.246 + 0.813i)15-s + (3.91 − 0.836i)16-s + (−1.26 + 0.384i)17-s + ⋯ |
| L(s) = 1 | + (−0.0526 − 0.998i)2-s + (0.248 − 0.0957i)3-s + (−0.994 + 0.105i)4-s + (−0.507 + 0.650i)5-s + (−0.108 − 0.242i)6-s + (0.446 − 1.24i)7-s + (0.157 + 0.987i)8-s + (−0.688 + 0.624i)9-s + (0.676 + 0.472i)10-s + (−0.284 − 1.26i)11-s + (−0.236 + 0.121i)12-s + (−1.66 + 0.460i)13-s + (−1.27 − 0.380i)14-s + (−0.0637 + 0.210i)15-s + (0.977 − 0.209i)16-s + (−0.307 + 0.0933i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112770 + 0.353010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112770 + 0.353010i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0744 + 1.41i)T \) |
| good | 3 | \( 1 + (-0.429 + 0.165i)T + (2.22 - 2.01i)T^{2} \) |
| 5 | \( 1 + (1.13 - 1.45i)T + (-1.21 - 4.85i)T^{2} \) |
| 7 | \( 1 + (-1.18 + 3.30i)T + (-5.41 - 4.44i)T^{2} \) |
| 11 | \( 1 + (0.943 + 4.20i)T + (-9.94 + 4.70i)T^{2} \) |
| 13 | \( 1 + (6.00 - 1.66i)T + (11.1 - 6.68i)T^{2} \) |
| 17 | \( 1 + (1.26 - 0.384i)T + (14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (2.34 + 4.65i)T + (-11.3 + 15.2i)T^{2} \) |
| 23 | \( 1 + (1.17 - 0.174i)T + (22.0 - 6.67i)T^{2} \) |
| 29 | \( 1 + (1.47 + 8.47i)T + (-27.3 + 9.76i)T^{2} \) |
| 31 | \( 1 + (5.20 - 7.78i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (0.939 - 0.810i)T + (5.42 - 36.5i)T^{2} \) |
| 41 | \( 1 + (-0.497 + 1.98i)T + (-36.1 - 19.3i)T^{2} \) |
| 43 | \( 1 + (-0.541 - 1.22i)T + (-28.8 + 31.8i)T^{2} \) |
| 47 | \( 1 + (-0.859 - 1.04i)T + (-9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (0.134 - 0.777i)T + (-49.9 - 17.8i)T^{2} \) |
| 59 | \( 1 + (-1.79 + 6.49i)T + (-50.6 - 30.3i)T^{2} \) |
| 61 | \( 1 + (-11.1 - 0.274i)T + (60.9 + 2.99i)T^{2} \) |
| 67 | \( 1 + (-11.2 + 10.6i)T + (3.28 - 66.9i)T^{2} \) |
| 71 | \( 1 + (7.58 - 0.372i)T + (70.6 - 6.95i)T^{2} \) |
| 73 | \( 1 + (-0.740 - 2.06i)T + (-56.4 + 46.3i)T^{2} \) |
| 79 | \( 1 + (2.22 - 0.218i)T + (77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (7.14 + 6.15i)T + (12.1 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.57 - 0.827i)T + (85.1 + 25.8i)T^{2} \) |
| 97 | \( 1 + (-3.38 + 0.674i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.78664344802435671789737838925, −9.673974774914584855115340319224, −8.585431358469325974197987690934, −7.79886723246103491460158310806, −7.00499032814371558908647613618, −5.30746059049677218947475253801, −4.33718075252412212463210305308, −3.25032233689322135343159667492, −2.24624895682007475434112934601, −0.20489047153276074519939892910,
2.34578482161543308107598881423, 4.05672412544174404340100786000, 5.07209260734179947984467309118, 5.68514450959298170021069719629, 7.04364932183445282955884956425, 7.930093247450943209027384179831, 8.621381475588435279667978293139, 9.363286558346877980911642246175, 10.13726973456164231123909568902, 11.74729766891760772898084786569
