| L(s) = 1 | + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)5-s + (0.766 − 0.642i)6-s + (−1.21 − 2.10i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.173 + 0.984i)10-s + (0.983 − 1.70i)11-s + (−0.499 − 0.866i)12-s + (3.82 − 3.20i)13-s + (−2.28 + 0.832i)14-s + (−0.939 − 0.342i)15-s + (0.766 + 0.642i)16-s + (0.779 − 4.42i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (−0.420 + 0.152i)5-s + (0.312 − 0.262i)6-s + (−0.460 − 0.797i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.0549 + 0.311i)10-s + (0.296 − 0.513i)11-s + (−0.144 − 0.249i)12-s + (1.05 − 0.889i)13-s + (−0.611 + 0.222i)14-s + (−0.242 − 0.0883i)15-s + (0.191 + 0.160i)16-s + (0.189 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.909171 - 1.11061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.909171 - 1.11061i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (2.14 + 3.79i)T \) |
| good | 7 | \( 1 + (1.21 + 2.10i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.983 + 1.70i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.82 + 3.20i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.779 + 4.42i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.62 - 0.590i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0965 - 0.547i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 41 | \( 1 + (-2.99 - 2.51i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.11 + 0.770i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.19 - 6.77i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.31 - 0.479i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.61 - 9.16i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.47 + 0.537i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.06 + 6.02i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.04 - 1.47i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.44 - 2.05i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (6.63 + 5.56i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.19 - 12.4i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.72 - 1.44i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.39 - 7.91i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.78240216096299351496675049222, −9.656743151267600189080311147637, −8.933812056527897071663033631009, −7.977651423049290177005902221266, −6.98727509877180995471243476996, −5.75024441015268786632817995978, −4.47191479196420545837881690053, −3.58667549515429416171158747024, −2.81126217182433215966674550997, −0.796468147375735600229551137879,
1.77044246298152107779159413850, 3.44628120371738924394252647069, 4.29866582654802398180111582887, 5.75282793940967130979456946851, 6.46661110610796748331432395982, 7.36378518336228202779086060160, 8.518163208686985524087124521642, 8.793480186771740924743888060157, 9.850635246829338957956463711371, 11.02987974123351676437251668340
