| L(s) = 1 | − 2.70·2-s + 3-s + 5.31·4-s − 2.83·5-s − 2.70·6-s + 2.54·7-s − 8.94·8-s + 9-s + 7.66·10-s − 11-s + 5.31·12-s − 3.40·13-s − 6.87·14-s − 2.83·15-s + 13.5·16-s + 3.34·17-s − 2.70·18-s + 0.886·19-s − 15.0·20-s + 2.54·21-s + 2.70·22-s − 2.73·23-s − 8.94·24-s + 3.03·25-s + 9.21·26-s + 27-s + 13.5·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 0.577·3-s + 2.65·4-s − 1.26·5-s − 1.10·6-s + 0.961·7-s − 3.16·8-s + 0.333·9-s + 2.42·10-s − 0.301·11-s + 1.53·12-s − 0.945·13-s − 1.83·14-s − 0.732·15-s + 3.39·16-s + 0.812·17-s − 0.637·18-s + 0.203·19-s − 3.36·20-s + 0.555·21-s + 0.576·22-s − 0.569·23-s − 1.82·24-s + 0.607·25-s + 1.80·26-s + 0.192·27-s + 2.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 + 2.83T + 5T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 0.886T + 19T^{2} \) |
| 23 | \( 1 + 2.73T + 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 - 8.50T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 + 6.27T + 47T^{2} \) |
| 53 | \( 1 + 2.19T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 8.61T + 73T^{2} \) |
| 79 | \( 1 + 0.270T + 79T^{2} \) |
| 83 | \( 1 + 8.98T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.564348947905921706127479810496, −7.982719090722110882384970964647, −7.61995944538656054828899527421, −7.13084850980729346221806569674, −5.83535727664250391335354166083, −4.58227061428859228082783916471, −3.38797945993740204332886293281, −2.41937786781301920624318239816, −1.36083932298764467977444883637, 0,
1.36083932298764467977444883637, 2.41937786781301920624318239816, 3.38797945993740204332886293281, 4.58227061428859228082783916471, 5.83535727664250391335354166083, 7.13084850980729346221806569674, 7.61995944538656054828899527421, 7.982719090722110882384970964647, 8.564348947905921706127479810496
