| L(s) = 1 | + (0.984 + 0.173i)2-s + (−1.43 − 0.974i)3-s + (0.939 + 0.342i)4-s + (1.35 − 1.77i)5-s + (−1.24 − 1.20i)6-s + (−1.28 + 0.739i)7-s + (0.866 + 0.5i)8-s + (1.10 + 2.79i)9-s + (1.64 − 1.51i)10-s + (2.81 + 1.62i)11-s + (−1.01 − 1.40i)12-s + (3.29 − 2.76i)13-s + (−1.38 + 0.505i)14-s + (−3.67 + 1.21i)15-s + (0.766 + 0.642i)16-s + (0.748 − 4.24i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (−0.826 − 0.562i)3-s + (0.469 + 0.171i)4-s + (0.607 − 0.794i)5-s + (−0.506 − 0.493i)6-s + (−0.483 + 0.279i)7-s + (0.306 + 0.176i)8-s + (0.367 + 0.930i)9-s + (0.520 − 0.478i)10-s + (0.849 + 0.490i)11-s + (−0.292 − 0.405i)12-s + (0.913 − 0.766i)13-s + (−0.371 + 0.135i)14-s + (−0.949 + 0.314i)15-s + (0.191 + 0.160i)16-s + (0.181 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70293 - 0.838770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70293 - 0.838770i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 3 | \( 1 + (1.43 + 0.974i)T \) |
| 5 | \( 1 + (-1.35 + 1.77i)T \) |
| 19 | \( 1 + (-1.45 + 4.10i)T \) |
| good | 7 | \( 1 + (1.28 - 0.739i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.81 - 1.62i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.29 + 2.76i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.748 + 4.24i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (7.23 + 2.63i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.791 - 4.48i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.19 + 1.26i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.11T + 37T^{2} \) |
| 41 | \( 1 + (1.02 + 0.862i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.81 + 7.73i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.06 - 11.6i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.503 - 1.38i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (2.10 - 11.9i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.53 - 3.10i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.31 - 13.1i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.07 - 0.757i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.62 + 9.08i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.55 + 5.43i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.29 + 2.23i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.44 + 2.89i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.63 - 9.26i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.77478030580128627382287194237, −9.820048498804390061167386029610, −8.856343134471542447970210265226, −7.71825404765691817469916536309, −6.64604494415281000572945696556, −5.98703045273034123395954798651, −5.19391705807394205796543541541, −4.22255743421841367172529815525, −2.56444848847842326354816395830, −1.09479866032038923317757131946,
1.65556570943739966142968683534, 3.58789533166274611305211252465, 3.93023192259009817502158199678, 5.51226269889424069393032007830, 6.38934735533636847113541768099, 6.51507153630478692281833823083, 8.165261679672271778033856599824, 9.673814369969194508022128083604, 10.00360168920113563887175447794, 11.03954968316784724599060894568
