| L(s) = 1 | − 2-s − 2.82·3-s + 4-s + 2.27·5-s + 2.82·6-s − 0.755·7-s − 8-s + 4.98·9-s − 2.27·10-s + 1.66·11-s − 2.82·12-s − 0.423·13-s + 0.755·14-s − 6.44·15-s + 16-s − 4.17·17-s − 4.98·18-s − 0.322·19-s + 2.27·20-s + 2.13·21-s − 1.66·22-s + 5.28·23-s + 2.82·24-s + 0.197·25-s + 0.423·26-s − 5.61·27-s − 0.755·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63·3-s + 0.5·4-s + 1.01·5-s + 1.15·6-s − 0.285·7-s − 0.353·8-s + 1.66·9-s − 0.720·10-s + 0.502·11-s − 0.815·12-s − 0.117·13-s + 0.201·14-s − 1.66·15-s + 0.250·16-s − 1.01·17-s − 1.17·18-s − 0.0739·19-s + 0.509·20-s + 0.466·21-s − 0.355·22-s + 1.10·23-s + 0.576·24-s + 0.0394·25-s + 0.0829·26-s − 1.08·27-s − 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 + 0.755T + 7T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 + 0.423T + 13T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 19 | \( 1 + 0.322T + 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 - 5.47T + 31T^{2} \) |
| 37 | \( 1 + 9.38T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 - 2.13T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 8.17T + 59T^{2} \) |
| 61 | \( 1 + 2.64T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 4.51T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 0.712T + 79T^{2} \) |
| 83 | \( 1 + 1.59T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 9.98T + 97T^{2} \) |
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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
−8.151955687486210088141519579011, −6.85827307536076834095261258331, −6.65633843336382848590829706126, −6.09530766427712446480862579717, −5.17109056568308957681840406022, −4.68729314290602838777802952509, −3.31307855018587652768000629258, −2.05461697648922034722262438659, −1.18840096835626391756027390877, 0,
1.18840096835626391756027390877, 2.05461697648922034722262438659, 3.31307855018587652768000629258, 4.68729314290602838777802952509, 5.17109056568308957681840406022, 6.09530766427712446480862579717, 6.65633843336382848590829706126, 6.85827307536076834095261258331, 8.151955687486210088141519579011
