L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (2.16 + 2.16i)5-s + (2.62 − 0.292i)7-s + (−0.707 − 0.707i)8-s + 3.06·10-s + (−0.516 − 0.516i)11-s + (3.60 − 0.0469i)13-s + (1.65 − 2.06i)14-s − 1.00·16-s + 2.57·17-s + (−3.82 − 3.82i)19-s + (2.16 − 2.16i)20-s − 0.729·22-s + 6.97i·23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.969 + 0.969i)5-s + (0.993 − 0.110i)7-s + (−0.250 − 0.250i)8-s + 0.969·10-s + (−0.155 − 0.155i)11-s + (0.999 − 0.0130i)13-s + (0.441 − 0.552i)14-s − 0.250·16-s + 0.624·17-s + (−0.876 − 0.876i)19-s + (0.484 − 0.484i)20-s − 0.155·22-s + 1.45i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.095780276\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.095780276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.292i)T \) |
| 13 | \( 1 + (-3.60 + 0.0469i)T \) |
good | 5 | \( 1 + (-2.16 - 2.16i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.516 + 0.516i)T + 11iT^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + (3.82 + 3.82i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.97iT - 23T^{2} \) |
| 29 | \( 1 - 6.32T + 29T^{2} \) |
| 31 | \( 1 + (4.33 + 4.33i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.58 + 3.58i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.176 - 0.176i)T + 41iT^{2} \) |
| 43 | \( 1 - 7.89iT - 43T^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 - 0.514T + 53T^{2} \) |
| 59 | \( 1 + (-3.17 + 3.17i)T - 59iT^{2} \) |
| 61 | \( 1 - 6.41iT - 61T^{2} \) |
| 67 | \( 1 + (3.96 - 3.96i)T - 67iT^{2} \) |
| 71 | \( 1 + (8.12 - 8.12i)T - 71iT^{2} \) |
| 73 | \( 1 + (-11.6 + 11.6i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + (1.01 + 1.01i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.10 - 1.10i)T - 89iT^{2} \) |
| 97 | \( 1 + (9.34 + 9.34i)T + 97iT^{2} \) |
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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
−9.528561099497377867768120226896, −8.642843017673834667601429270089, −7.68775787061629486859824980890, −6.74310735141473622457574831496, −5.94829166337836944862768443498, −5.30831938688729364967213475602, −4.23719564627788745170958747156, −3.22783872609124218754817339725, −2.28268620356781601687882686831, −1.34154451992563246736847127679,
1.27656675500465587345892387259, 2.24358506909379042911250269014, 3.72326657862004871183007870949, 4.71069893355661087343975287488, 5.28732270263829400550716997944, 6.04162264997037301496315685946, 6.81978737057573794188637936413, 8.122841957070079797986989956861, 8.449444465263725818423021401038, 9.141260997484540057187342035250