L(s) = 1 | − 6.80i·2-s + (7.51 + 4.95i)3-s − 30.3·4-s − 9.67i·5-s + (33.7 − 51.1i)6-s − 60.1·7-s + 97.5i·8-s + (31.9 + 74.4i)9-s − 65.8·10-s + 66.8i·11-s + (−227. − 150. i)12-s − 79.5·13-s + 409. i·14-s + (47.9 − 72.6i)15-s + 178.·16-s − 76.6i·17-s + ⋯ |
L(s) = 1 | − 1.70i·2-s + (0.834 + 0.550i)3-s − 1.89·4-s − 0.386i·5-s + (0.936 − 1.42i)6-s − 1.22·7-s + 1.52i·8-s + (0.394 + 0.919i)9-s − 0.658·10-s + 0.552i·11-s + (−1.58 − 1.04i)12-s − 0.471·13-s + 2.08i·14-s + (0.212 − 0.323i)15-s + 0.698·16-s − 0.265i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4396035020\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4396035020\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-7.51 - 4.95i)T \) |
| 59 | \( 1 + 453. iT \) |
good | 2 | \( 1 + 6.80iT - 16T^{2} \) |
| 5 | \( 1 + 9.67iT - 625T^{2} \) |
| 7 | \( 1 + 60.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 66.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 79.5T + 2.85e4T^{2} \) |
| 17 | \( 1 + 76.6iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 369.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 853. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 694. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.16e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.06e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 738. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.78e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.01e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.78e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 3.10e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.62e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.08e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.72e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.30e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.19e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.69e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.29e4T + 8.85e7T^{2} \) |
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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
−12.35393911306922304039846944172, −10.98590284514888591822728811270, −10.13435842692705232747530215517, −9.401239376035001101147638151223, −8.816678118328668013860580542103, −7.18606127837571120953884099243, −5.09600999289748753804166200581, −3.90255039991792144641504341175, −3.03308525667817380550579508209, −1.81048834442255673303466452444,
0.13930142720580375428466328384, 2.78060267792106646282715519283, 4.22982713273580723581070678547, 6.04890917853507185187729798869, 6.63428874767572488091731011716, 7.51214046044926842070210067117, 8.592640081960887661905963860272, 9.257593300494376075784405317279, 10.51208284557347584248511090528, 12.49416321266634703555404001622