L(s) = 1 | + (−0.5 − 0.866i)2-s + (−1.30 − 2.25i)3-s + (0.500 − 0.866i)4-s + 3.76·5-s + (−1.30 + 2.25i)6-s + (−0.0835 + 0.144i)7-s − 3·8-s + (−1.88 + 3.26i)9-s + (−1.88 − 3.26i)10-s + (−0.5 − 0.866i)11-s − 2.60·12-s + (0.199 + 3.60i)13-s + 0.167·14-s + (−4.90 − 8.49i)15-s + (0.500 + 0.866i)16-s + (−2.18 + 3.78i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.751 − 1.30i)3-s + (0.250 − 0.433i)4-s + 1.68·5-s + (−0.531 + 0.919i)6-s + (−0.0315 + 0.0546i)7-s − 1.06·8-s + (−0.628 + 1.08i)9-s + (−0.595 − 1.03i)10-s + (−0.150 − 0.261i)11-s − 0.751·12-s + (0.0552 + 0.998i)13-s + 0.0446·14-s + (−1.26 − 2.19i)15-s + (0.125 + 0.216i)16-s + (−0.529 + 0.917i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.393405 - 0.872638i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.393405 - 0.872638i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.199 - 3.60i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.30 + 2.25i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 7 | \( 1 + (0.0835 - 0.144i)T + (-3.5 - 6.06i)T^{2} \) |
| 17 | \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.68 + 2.91i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.60 - 4.50i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.967 + 1.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.86T + 31T^{2} \) |
| 37 | \( 1 + (4.31 + 7.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.633 - 1.09i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.60 + 6.23i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 + (1.06 - 1.85i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.40 + 4.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.06 - 3.58i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + (7.15 + 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.78 - 3.08i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.72826992280505370952037309872, −11.62569672545782006515828101109, −10.87246515019325230800700496814, −9.783009441604310892954384095437, −8.876701740028882947880482124057, −6.96642708192408383647240671901, −6.21302881003419412178982298768, −5.45577233419780419919168320201, −2.33448561716203863438677919592, −1.38335798203332353205906776523,
2.91724849921526234515690585624, 4.90221327232450364826769034620, 5.81053370930250846943827634553, 6.77831233247347551298271155952, 8.453062993089041897617818191866, 9.615600084736281348359636081597, 10.13568424336519535674425207219, 11.18654381735599218073917420652, 12.44717552100503900270911124591, 13.53990537373612507181701169113