| L(s) = 1 | + (−0.988 + 0.149i)3-s + (−0.540 + 1.37i)5-s + (−0.639 − 2.56i)7-s + (0.955 − 0.294i)9-s + (2.63 + 0.812i)11-s + (0.155 + 0.680i)13-s + (0.329 − 1.44i)15-s + (1.79 − 1.22i)17-s + (−0.272 − 0.472i)19-s + (1.01 + 2.44i)21-s + (3.73 + 2.54i)23-s + (2.05 + 1.90i)25-s + (−0.900 + 0.433i)27-s + (8.31 + 4.00i)29-s + (3.06 − 5.30i)31-s + ⋯ |
| L(s) = 1 | + (−0.570 + 0.0860i)3-s + (−0.241 + 0.616i)5-s + (−0.241 − 0.970i)7-s + (0.318 − 0.0982i)9-s + (0.794 + 0.244i)11-s + (0.0430 + 0.188i)13-s + (0.0850 − 0.372i)15-s + (0.436 − 0.297i)17-s + (−0.0625 − 0.108i)19-s + (0.221 + 0.533i)21-s + (0.778 + 0.531i)23-s + (0.411 + 0.381i)25-s + (−0.173 + 0.0835i)27-s + (1.54 + 0.743i)29-s + (0.549 − 0.952i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19862 + 0.139439i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19862 + 0.139439i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.988 - 0.149i)T \) |
| 7 | \( 1 + (0.639 + 2.56i)T \) |
| good | 5 | \( 1 + (0.540 - 1.37i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-2.63 - 0.812i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.155 - 0.680i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-1.79 + 1.22i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (0.272 + 0.472i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.73 - 2.54i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-8.31 - 4.00i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-3.06 + 5.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.421 - 5.62i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-1.13 - 1.42i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (0.737 - 0.924i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-5.12 + 4.75i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.353 + 4.71i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (0.551 + 1.40i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.912 - 12.1i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-6.90 + 11.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.0 + 5.32i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (10.7 + 9.94i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-0.506 - 0.877i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.62 - 7.14i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (2.47 - 0.762i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 9.92T + 97T^{2} \) |
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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
−10.73660657308773164596616482911, −10.02557543307606775207597916466, −9.166625088564076665373501280220, −7.87085874229074357309344978717, −6.91064613349812933349618883716, −6.50851441392487347119920431876, −5.08980645769251693617277102080, −4.09643071528839197196683439366, −3.07294473846373439863801486225, −1.09720660537255964474260536125,
1.01705590995554047817430755702, 2.74124289196027044468554391061, 4.17766351843787601376751960721, 5.17425938964627784190035067933, 6.08096537300910538707568925101, 6.87328566898957816937887712072, 8.255256095662152055559782636883, 8.797693031452427450517983702254, 9.770367293818742094055052657206, 10.73635300417959505686605589715
