L(s) = 1 | + (1.29 − 2.24i)2-s + (−0.259 − 0.449i)3-s + (−2.35 − 4.07i)4-s + (0.806 − 1.39i)5-s − 1.34·6-s + (−2.13 − 1.56i)7-s − 6.99·8-s + (1.36 − 2.36i)9-s + (−2.08 − 3.61i)10-s + (1.35 + 2.34i)11-s + (−1.21 + 2.11i)12-s + (−6.27 + 2.74i)14-s − 0.835·15-s + (−4.34 + 7.53i)16-s + (−1.56 − 2.70i)17-s + (−3.53 − 6.12i)18-s + ⋯ |
L(s) = 1 | + (0.915 − 1.58i)2-s + (−0.149 − 0.259i)3-s + (−1.17 − 2.03i)4-s + (0.360 − 0.624i)5-s − 0.547·6-s + (−0.805 − 0.592i)7-s − 2.47·8-s + (0.455 − 0.788i)9-s + (−0.659 − 1.14i)10-s + (0.407 + 0.706i)11-s + (−0.351 + 0.609i)12-s + (−1.67 + 0.734i)14-s − 0.215·15-s + (−1.08 + 1.88i)16-s + (−0.379 − 0.656i)17-s + (−0.833 − 1.44i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.063019531\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.063019531\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.13 + 1.56i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.29 + 2.24i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.259 + 0.449i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.806 + 1.39i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.35 - 2.34i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.56 + 2.70i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.84 + 3.18i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.993 - 1.71i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + (-5.23 - 9.07i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.97 + 5.15i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 + (-0.527 + 0.913i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.63 + 6.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.71 - 9.89i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.46 + 2.53i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.79 + 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 + (-4.55 - 7.88i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 + (-0.879 + 1.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−9.420959083072459513975170264359, −9.118798291348273802590942659797, −7.28645616699679557835748285240, −6.53043791091377700196024390677, −5.46947786930391089726825034409, −4.60991557344420220507448212310, −3.83961696628356078387323010502, −2.92362086219957227438350503518, −1.63420438444421545188188892238, −0.69105622289586672664483522891,
2.52384537505261465767407251119, 3.65923091541204209077222528671, 4.43994862474449808252232384355, 5.58590237280395326341211049941, 6.11592191124507441424190215581, 6.65410322607046301228489395066, 7.70697232289672876680702134880, 8.325741658498016423659631995002, 9.347312425633898528488821283384, 10.11735162417603551808062081729