| L(s) = 1 | − 2-s + 0.421·3-s + 4-s − 0.176·5-s − 0.421·6-s + 0.315·7-s − 8-s − 2.82·9-s + 0.176·10-s + 2.09·11-s + 0.421·12-s + 3.82·13-s − 0.315·14-s − 0.0745·15-s + 16-s + 4.64·17-s + 2.82·18-s − 2.28·19-s − 0.176·20-s + 0.133·21-s − 2.09·22-s − 3.96·23-s − 0.421·24-s − 4.96·25-s − 3.82·26-s − 2.45·27-s + 0.315·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.243·3-s + 0.5·4-s − 0.0790·5-s − 0.172·6-s + 0.119·7-s − 0.353·8-s − 0.940·9-s + 0.0558·10-s + 0.630·11-s + 0.121·12-s + 1.06·13-s − 0.0844·14-s − 0.0192·15-s + 0.250·16-s + 1.12·17-s + 0.665·18-s − 0.524·19-s − 0.0395·20-s + 0.0290·21-s − 0.445·22-s − 0.827·23-s − 0.0860·24-s − 0.993·25-s − 0.750·26-s − 0.472·27-s + 0.0597·28-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 | \( 1 - 0.421T + 3T^{2} \) |
| 5 | \( 1 + 0.176T + 5T^{2} \) |
| 7 | \( 1 - 0.315T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 4.64T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 - 4.58T + 43T^{2} \) |
| 47 | \( 1 + 5.95T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.57T + 61T^{2} \) |
| 67 | \( 1 - 6.03T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 5.13T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−8.165369392103001246679074561234, −7.60718236460652606293491534474, −6.66502121373292305049541626932, −5.87450056987824685154313961812, −5.40855783228535047707133084763, −3.84201431816962992022046944666, −3.53944048061539856022222990242, −2.26020846308550163929598159873, −1.41938792008252632248417759033, 0,
1.41938792008252632248417759033, 2.26020846308550163929598159873, 3.53944048061539856022222990242, 3.84201431816962992022046944666, 5.40855783228535047707133084763, 5.87450056987824685154313961812, 6.66502121373292305049541626932, 7.60718236460652606293491534474, 8.165369392103001246679074561234
