L(s) = 1 | − 2-s + 3-s + 4-s − 1.66·5-s − 6-s − 2.43·7-s − 8-s + 9-s + 1.66·10-s − 11-s + 12-s + 6.41·13-s + 2.43·14-s − 1.66·15-s + 16-s − 0.265·17-s − 18-s − 4.82·19-s − 1.66·20-s − 2.43·21-s + 22-s + 5.37·23-s − 24-s − 2.22·25-s − 6.41·26-s + 27-s − 2.43·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.744·5-s − 0.408·6-s − 0.920·7-s − 0.353·8-s + 0.333·9-s + 0.526·10-s − 0.301·11-s + 0.288·12-s + 1.78·13-s + 0.650·14-s − 0.429·15-s + 0.250·16-s − 0.0644·17-s − 0.235·18-s − 1.10·19-s − 0.372·20-s − 0.531·21-s + 0.213·22-s + 1.12·23-s − 0.204·24-s − 0.445·25-s − 1.25·26-s + 0.192·27-s − 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 13 | \( 1 - 6.41T + 13T^{2} \) |
| 17 | \( 1 + 0.265T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + 6.27T + 37T^{2} \) |
| 41 | \( 1 - 7.91T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 0.474T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 67 | \( 1 - 1.15T + 67T^{2} \) |
| 71 | \( 1 - 6.34T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 - 4.10T + 89T^{2} \) |
| 97 | \( 1 - 4.25T + 97T^{2} \) |
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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
−8.315583506182269107091838909957, −7.50371880921043911309068608823, −6.65095096406869249314131224096, −6.25118974131979911294601790326, −5.07511229753453376453874336349, −3.80842564834959520680796714861, −3.50514389383045682347497643772, −2.47802105401937259561813189382, −1.30257896553921120174297741290, 0,
1.30257896553921120174297741290, 2.47802105401937259561813189382, 3.50514389383045682347497643772, 3.80842564834959520680796714861, 5.07511229753453376453874336349, 6.25118974131979911294601790326, 6.65095096406869249314131224096, 7.50371880921043911309068608823, 8.315583506182269107091838909957