| L(s) = 1 | + (0.766 + 0.642i)2-s + (0.654 + 1.60i)3-s + (0.173 + 0.984i)4-s + (0.190 + 1.08i)5-s + (−0.529 + 1.64i)6-s + (2.53 − 0.751i)7-s + (−0.500 + 0.866i)8-s + (−2.14 + 2.09i)9-s + (−0.549 + 0.951i)10-s + (0.0294 − 0.166i)11-s + (−1.46 + 0.922i)12-s + (−0.287 − 1.63i)13-s + (2.42 + 1.05i)14-s + (−1.61 + 1.01i)15-s + (−0.939 + 0.342i)16-s + (0.236 − 0.409i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.377 + 0.925i)3-s + (0.0868 + 0.492i)4-s + (0.0853 + 0.484i)5-s + (−0.216 + 0.673i)6-s + (0.958 − 0.284i)7-s + (−0.176 + 0.306i)8-s + (−0.714 + 0.699i)9-s + (−0.173 + 0.301i)10-s + (0.00887 − 0.0503i)11-s + (−0.423 + 0.266i)12-s + (−0.0797 − 0.452i)13-s + (0.648 + 0.281i)14-s + (−0.415 + 0.261i)15-s + (−0.234 + 0.0855i)16-s + (0.0573 − 0.0992i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31139 + 1.66061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31139 + 1.66061i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.654 - 1.60i)T \) |
| 7 | \( 1 + (-2.53 + 0.751i)T \) |
| good | 5 | \( 1 + (-0.190 - 1.08i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.0294 + 0.166i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (0.287 + 1.63i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.236 + 0.409i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.72 + 2.98i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.39 - 2.00i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.0779 - 0.442i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.131 + 0.745i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 + (0.223 + 1.26i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.12 - 3.46i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.836 + 4.74i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.65 - 2.87i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.62 - 1.31i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.36 + 13.3i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.2 + 8.63i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.29 - 10.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + (-1.86 - 1.56i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.32 - 13.1i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (0.437 + 0.757i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.50 - 2.94i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.37637738496304269178165502878, −10.82815193837165380343519807983, −9.868436163187087616385720345877, −8.695598205434279905282154289414, −7.947847429894828134730674006675, −6.91258413735738068763995540283, −5.57158141576988455992960336721, −4.73417004050267003611386373403, −3.72054977239115800660680733252, −2.47790961670094481977251814800,
1.38967009288812998420548524392, 2.44685643713525150111463404739, 4.00128524869815444599149182086, 5.19110400358596538177870791462, 6.19339221905317136740526466137, 7.34975959791019726029074625178, 8.388821802812574692912956833151, 9.060243494433891831909094736159, 10.36716737603892564163765029364, 11.43554581600339030738998880918
