| L(s) = 1 | + 1.60·2-s + 3-s + 0.575·4-s + 1.01·5-s + 1.60·6-s − 7-s − 2.28·8-s + 9-s + 1.63·10-s + 3.71·11-s + 0.575·12-s − 1.76·13-s − 1.60·14-s + 1.01·15-s − 4.81·16-s + 0.368·17-s + 1.60·18-s − 3.48·19-s + 0.585·20-s − 21-s + 5.96·22-s − 1.25·23-s − 2.28·24-s − 3.96·25-s − 2.83·26-s + 27-s − 0.575·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 0.577·3-s + 0.287·4-s + 0.454·5-s + 0.655·6-s − 0.377·7-s − 0.808·8-s + 0.333·9-s + 0.516·10-s + 1.12·11-s + 0.166·12-s − 0.490·13-s − 0.428·14-s + 0.262·15-s − 1.20·16-s + 0.0894·17-s + 0.378·18-s − 0.799·19-s + 0.130·20-s − 0.218·21-s + 1.27·22-s − 0.261·23-s − 0.466·24-s − 0.792·25-s − 0.556·26-s + 0.192·27-s − 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 1.60T + 2T^{2} \) |
| 5 | \( 1 - 1.01T + 5T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 0.368T + 17T^{2} \) |
| 19 | \( 1 + 3.48T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 + 5.77T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 1.69T + 43T^{2} \) |
| 47 | \( 1 + 4.72T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 3.31T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 3.18T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 - 9.68T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 0.483T + 89T^{2} \) |
| 97 | \( 1 + 2.08T + 97T^{2} \) |
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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
−7.21365678634039722560080294002, −6.71131813493750341617709003096, −5.91243808280135150234248842264, −5.44246081674298588038810843325, −4.49986333410013821648271582007, −3.83419976839674727051140996519, −3.41749080162268999876264024432, −2.36994844966670688382387528427, −1.69391889064648258862389308197, 0,
1.69391889064648258862389308197, 2.36994844966670688382387528427, 3.41749080162268999876264024432, 3.83419976839674727051140996519, 4.49986333410013821648271582007, 5.44246081674298588038810843325, 5.91243808280135150234248842264, 6.71131813493750341617709003096, 7.21365678634039722560080294002
