| L(s) = 1 | + (−0.590 − 1.47i)3-s + (−0.463 + 1.91i)5-s + (−2.06 + 1.65i)7-s + (0.342 − 0.327i)9-s + (2.19 + 0.104i)11-s + (−4.70 − 4.07i)13-s + (3.09 − 0.444i)15-s + (3.77 − 5.30i)17-s + (−3.38 − 4.75i)19-s + (3.65 + 2.07i)21-s + (−1.61 − 4.51i)23-s + (1.00 + 0.518i)25-s + (−5.02 − 2.29i)27-s + (−0.604 − 1.32i)29-s + (0.506 + 0.644i)31-s + ⋯ |
| L(s) = 1 | + (−0.341 − 0.851i)3-s + (−0.207 + 0.854i)5-s + (−0.780 + 0.624i)7-s + (0.114 − 0.109i)9-s + (0.660 + 0.0314i)11-s + (−1.30 − 1.13i)13-s + (0.798 − 0.114i)15-s + (0.916 − 1.28i)17-s + (−0.776 − 1.08i)19-s + (0.798 + 0.452i)21-s + (−0.336 − 0.941i)23-s + (0.201 + 0.103i)25-s + (−0.966 − 0.441i)27-s + (−0.112 − 0.245i)29-s + (0.0910 + 0.115i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.375323 - 0.683060i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375323 - 0.683060i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.06 - 1.65i)T \) |
| 23 | \( 1 + (1.61 + 4.51i)T \) |
| good | 3 | \( 1 + (0.590 + 1.47i)T + (-2.17 + 2.07i)T^{2} \) |
| 5 | \( 1 + (0.463 - 1.91i)T + (-4.44 - 2.29i)T^{2} \) |
| 11 | \( 1 + (-2.19 - 0.104i)T + (10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (4.70 + 4.07i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.77 + 5.30i)T + (-5.56 - 16.0i)T^{2} \) |
| 19 | \( 1 + (3.38 + 4.75i)T + (-6.21 + 17.9i)T^{2} \) |
| 29 | \( 1 + (0.604 + 1.32i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.506 - 0.644i)T + (-7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (0.000178 + 0.000187i)T + (-1.76 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-1.97 - 6.74i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (4.68 + 0.674i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (8.36 + 4.82i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.259 - 0.0898i)T + (41.6 + 32.7i)T^{2} \) |
| 59 | \( 1 + (-0.731 + 3.79i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (-12.6 - 5.05i)T + (44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (-0.0709 + 0.137i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (2.79 - 1.79i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.179 + 1.88i)T + (-71.6 + 13.8i)T^{2} \) |
| 79 | \( 1 + (-1.37 + 0.477i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (5.85 + 1.71i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (12.5 + 9.87i)T + (20.9 + 86.4i)T^{2} \) |
| 97 | \( 1 + (-15.8 + 4.64i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.10530024631606070483229943329, −9.642306697944432258525978819172, −8.411826138950776909621825824479, −7.23495443389704221395216214456, −6.87820866447704104135309597185, −5.99892073217864490200210266724, −4.85564549744192883394424230891, −3.24341407820604780214417776606, −2.45068407884197487504793173702, −0.43891845936132500316040409850,
1.64958787626779242140218085868, 3.73772147597149821120728632249, 4.21933310692215258592786922334, 5.22972809341160300460451679149, 6.30387762685363664484393286774, 7.33867270483706464078294465265, 8.337373768180293123813452934868, 9.453183089059879917277005438724, 9.897442950110456235647771030138, 10.63649966077313443591392985495
