L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−3 + 5.19i)11-s + (0.499 + 0.866i)12-s + 4·13-s + (−0.500 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (0.499 − 0.866i)18-s + (2 + 3.46i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.408·6-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.904 + 1.56i)11-s + (0.144 + 0.249i)12-s + 1.10·13-s + (−0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (0.117 − 0.204i)18-s + (0.458 + 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.274527333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.274527333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.5 - 6.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−10.02915472423281967718623095507, −9.344241720767780230050374003544, −8.316436522520288248382299247245, −7.57282998070885332851231701866, −6.87795682516827910085335530861, −6.12245118870547359623758704778, −5.15168049654872499107074541254, −3.96829522069206310759225894935, −3.14349193830380660051388942909, −1.71435483724229543837037647339,
0.47835544750354221821814505255, 2.49890856179310639498412766204, 3.23206088956383991994552992044, 4.01698143673249771397461151957, 5.38608171809325907348819554556, 5.85703934004524287801330833319, 7.00426958437910036326627033484, 8.299811556066453435077801908893, 8.982693985890315287691356027468, 9.575662568854598870404956912236