| L(s) = 1 | + (−0.913 − 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (0.171 − 1.63i)5-s + (0.809 + 0.587i)6-s + (2.59 + 0.512i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.822 + 1.42i)10-s + (3.21 − 0.833i)11-s + (−0.499 − 0.866i)12-s + (−5.56 + 4.04i)13-s + (−2.16 − 1.52i)14-s + (−0.508 + 1.56i)15-s + (−0.104 + 0.994i)16-s + (3.87 − 1.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 0.287i)2-s + (−0.564 − 0.120i)3-s + (0.334 + 0.371i)4-s + (0.0769 − 0.731i)5-s + (0.330 + 0.239i)6-s + (0.981 + 0.193i)7-s + (−0.109 − 0.336i)8-s + (0.304 + 0.135i)9-s + (−0.260 + 0.450i)10-s + (0.967 − 0.251i)11-s + (−0.144 − 0.249i)12-s + (−1.54 + 1.12i)13-s + (−0.578 − 0.407i)14-s + (−0.131 + 0.404i)15-s + (−0.0261 + 0.248i)16-s + (0.939 − 0.418i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.871380 - 0.459315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.871380 - 0.459315i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.913 + 0.406i)T \) |
| 3 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (-2.59 - 0.512i)T \) |
| 11 | \( 1 + (-3.21 + 0.833i)T \) |
| good | 5 | \( 1 + (-0.171 + 1.63i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (5.56 - 4.04i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.87 + 1.72i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.08 + 1.20i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.325 - 0.563i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.81 + 8.67i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.745 + 7.09i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-11.2 + 2.38i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-1.12 - 3.45i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.34T + 43T^{2} \) |
| 47 | \( 1 + (0.820 - 0.911i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (0.588 + 5.59i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (3.53 + 3.92i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.173 - 1.65i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-1.35 + 2.35i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.98 + 5.80i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.68 - 9.64i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-0.977 - 0.435i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-5.91 - 4.29i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (1.61 + 2.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.15 - 4.47i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.17447835731988378591312784235, −9.671984715508360817985537453763, −9.439776979120111798350077280802, −8.186390254331666790412669433679, −7.44060377817963775923930591143, −6.31822313221960461366324656614, −5.07122157242401012033215209342, −4.27190202050436858937914170285, −2.29308055592056999098745367920, −0.989630614266286933687188649862,
1.32357598055384996053584904611, 3.03161722241492391499768205738, 4.70441792028796792373745741442, 5.57411873785708099875690295853, 6.79016020427083041819204148736, 7.43855098363955027162567655138, 8.355848676764577698390535082219, 9.589259678380880996831281722611, 10.37932824920123732950127496731, 10.87697292789139155191589904252
