| L(s) = 1 | − 2-s − 1.40·3-s + 4-s − 1.65·5-s + 1.40·6-s + 2.90·7-s − 8-s − 1.03·9-s + 1.65·10-s − 1.47·11-s − 1.40·12-s + 1.93·13-s − 2.90·14-s + 2.31·15-s + 16-s + 1.62·17-s + 1.03·18-s − 4.58·19-s − 1.65·20-s − 4.06·21-s + 1.47·22-s + 7.27·23-s + 1.40·24-s − 2.27·25-s − 1.93·26-s + 5.65·27-s + 2.90·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.808·3-s + 0.5·4-s − 0.738·5-s + 0.571·6-s + 1.09·7-s − 0.353·8-s − 0.346·9-s + 0.521·10-s − 0.445·11-s − 0.404·12-s + 0.536·13-s − 0.776·14-s + 0.596·15-s + 0.250·16-s + 0.394·17-s + 0.245·18-s − 1.05·19-s − 0.369·20-s − 0.888·21-s + 0.315·22-s + 1.51·23-s + 0.285·24-s − 0.455·25-s − 0.379·26-s + 1.08·27-s + 0.549·28-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 | \( 1 + 1.40T + 3T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 - 1.93T + 13T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 - 7.27T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 + 0.384T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 9.12T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 1.89T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 0.0708T + 79T^{2} \) |
| 83 | \( 1 + 7.35T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 2.36T + 97T^{2} \) |
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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
−8.085203836367356539711044898060, −7.49917112577453850767921320784, −6.74223857977675211280740808390, −5.79708699661097699426632161868, −5.24102263563068691877719577804, −4.36914048856826141511640362474, −3.39430815224865348279840881445, −2.24779738039751620381577496616, −1.12637114502881848925056754695, 0,
1.12637114502881848925056754695, 2.24779738039751620381577496616, 3.39430815224865348279840881445, 4.36914048856826141511640362474, 5.24102263563068691877719577804, 5.79708699661097699426632161868, 6.74223857977675211280740808390, 7.49917112577453850767921320784, 8.085203836367356539711044898060
