| L(s) = 1 | + (−0.133 − 0.158i)2-s + (−1.17 + 1.27i)3-s + (0.339 − 1.92i)4-s + (−0.452 − 2.18i)5-s + (0.359 + 0.0165i)6-s + (2.83 − 0.500i)7-s + (−0.711 + 0.410i)8-s + (−0.246 − 2.98i)9-s + (−0.287 + 0.364i)10-s + (1.38 + 0.502i)11-s + (2.05 + 2.69i)12-s + (1.55 − 1.85i)13-s + (−0.457 − 0.384i)14-s + (3.32 + 1.99i)15-s + (−3.51 − 1.28i)16-s + (−1.21 − 0.704i)17-s + ⋯ |
| L(s) = 1 | + (−0.0943 − 0.112i)2-s + (−0.677 + 0.735i)3-s + (0.169 − 0.963i)4-s + (−0.202 − 0.979i)5-s + (0.146 + 0.00676i)6-s + (1.07 − 0.188i)7-s + (−0.251 + 0.145i)8-s + (−0.0822 − 0.996i)9-s + (−0.0910 + 0.115i)10-s + (0.416 + 0.151i)11-s + (0.593 + 0.777i)12-s + (0.431 − 0.514i)13-s + (−0.122 − 0.102i)14-s + (0.857 + 0.514i)15-s + (−0.879 − 0.320i)16-s + (−0.295 − 0.170i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.820993 - 0.408552i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.820993 - 0.408552i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.17 - 1.27i)T \) |
| 5 | \( 1 + (0.452 + 2.18i)T \) |
| good | 2 | \( 1 + (0.133 + 0.158i)T + (-0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-2.83 + 0.500i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.38 - 0.502i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.55 + 1.85i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.21 + 0.704i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.34 - 4.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 + 0.417i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.73 - 5.65i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.00 + 5.72i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-7.57 - 4.37i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.32 - 6.98i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.63 - 7.23i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.68 + 1.17i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 5.43iT - 53T^{2} \) |
| 59 | \( 1 + (-6.83 + 2.48i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.03 + 5.89i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.95 - 5.90i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.51 + 7.82i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (10.8 - 6.28i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.14 - 0.963i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.46 - 8.89i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (5.96 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.92 + 5.28i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−12.95823303995336679256007893678, −11.62442137866452795439326237739, −11.20902141372506019647859996749, −10.00473964523562505124424324986, −9.188301065720037510459117546086, −7.900222513844345046099270131744, −6.07946848337314648231171998763, −5.18499092112920053600816154097, −4.19629391223833928706121876132, −1.24410482425558575103496493421,
2.30377626966949057597555847370, 4.14328086447869570067467529664, 5.92456812069029420372314670012, 7.09711916299624062102349984031, 7.73488977890195316324696672290, 8.921205990131790808718722278018, 10.84289595041169572357254144950, 11.45756554096378291856769665060, 12.06658428134771943150148027014, 13.37390272207896748804198047798
