L(s) = 1 | + (−0.984 − 0.173i)2-s + (−1.64 − 0.541i)3-s + (0.939 + 0.342i)4-s + (2.21 + 0.302i)5-s + (1.52 + 0.819i)6-s + (−2.76 + 1.59i)7-s + (−0.866 − 0.5i)8-s + (2.41 + 1.78i)9-s + (−2.12 − 0.682i)10-s + (−1.63 − 0.942i)11-s + (−1.36 − 1.07i)12-s + (−1.97 + 1.65i)13-s + (3.00 − 1.09i)14-s + (−3.48 − 1.69i)15-s + (0.766 + 0.642i)16-s + (0.781 − 4.43i)17-s + ⋯ |
L(s) = 1 | + (−0.696 − 0.122i)2-s + (−0.949 − 0.312i)3-s + (0.469 + 0.171i)4-s + (0.990 + 0.135i)5-s + (0.623 + 0.334i)6-s + (−1.04 + 0.603i)7-s + (−0.306 − 0.176i)8-s + (0.804 + 0.594i)9-s + (−0.673 − 0.215i)10-s + (−0.492 − 0.284i)11-s + (−0.392 − 0.309i)12-s + (−0.546 + 0.458i)13-s + (0.801 − 0.291i)14-s + (−0.898 − 0.438i)15-s + (0.191 + 0.160i)16-s + (0.189 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00412967 + 0.0351312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00412967 + 0.0351312i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.64 + 0.541i)T \) |
| 5 | \( 1 + (-2.21 - 0.302i)T \) |
| 19 | \( 1 + (3.91 + 1.91i)T \) |
good | 7 | \( 1 + (2.76 - 1.59i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.63 + 0.942i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.97 - 1.65i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.781 + 4.43i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.269 - 0.0979i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.186 - 1.05i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (6.53 - 3.77i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.87T + 37T^{2} \) |
| 41 | \( 1 + (6.45 + 5.41i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.955 + 2.62i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.01 - 5.78i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (3.28 - 9.01i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.980 + 5.55i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (11.2 + 4.09i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.624 - 3.54i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (9.79 - 3.56i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (3.19 - 3.81i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (0.568 - 0.678i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.89 - 6.74i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-12.5 + 10.5i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.803 - 4.55i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.88160240932039681603223755334, −10.32787053479015271689406588274, −9.444774357293750044136840541767, −8.831556523921480456398247819268, −7.28077853799409165636514971306, −6.70053006634275693461347740888, −5.82039467710917030838622298930, −4.98699885849456338179066958418, −3.00031998677232854533573153009, −1.88654790083867303248093482774,
0.02632907952067336040804385710, 1.80599763886958433365447735488, 3.51167688223929831041038843826, 4.95334165272851290181612376144, 5.95861788503081220775080932392, 6.55253750001386643968282922656, 7.50799345598089029239836312103, 8.799546278303148353171063199505, 9.788888715572373130359841472747, 10.30503219028520630298636297809