| L(s) = 1 | + 2-s − 2.91·3-s + 4-s + 3.14·5-s − 2.91·6-s − 3.38·7-s + 8-s + 5.48·9-s + 3.14·10-s − 3.13·11-s − 2.91·12-s − 2.60·13-s − 3.38·14-s − 9.14·15-s + 16-s + 0.366·17-s + 5.48·18-s + 6.14·19-s + 3.14·20-s + 9.84·21-s − 3.13·22-s + 1.69·23-s − 2.91·24-s + 4.86·25-s − 2.60·26-s − 7.23·27-s − 3.38·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.68·3-s + 0.5·4-s + 1.40·5-s − 1.18·6-s − 1.27·7-s + 0.353·8-s + 1.82·9-s + 0.993·10-s − 0.945·11-s − 0.840·12-s − 0.723·13-s − 0.903·14-s − 2.36·15-s + 0.250·16-s + 0.0888·17-s + 1.29·18-s + 1.41·19-s + 0.702·20-s + 2.14·21-s − 0.668·22-s + 0.353·23-s − 0.594·24-s + 0.973·25-s − 0.511·26-s − 1.39·27-s − 0.638·28-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 | \( 1 + 2.91T + 3T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 + 3.13T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 - 0.366T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 - 0.436T + 29T^{2} \) |
| 31 | \( 1 + 0.0906T + 31T^{2} \) |
| 37 | \( 1 + 0.999T + 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 4.50T + 73T^{2} \) |
| 79 | \( 1 - 2.60T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 - 8.46T + 89T^{2} \) |
| 97 | \( 1 - 6.44T + 97T^{2} \) |
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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
−7.67483948332540496550750895946, −6.83201829495193969412646199372, −6.48700258312266337684566136984, −5.60673425928209812679721395124, −5.37707931082636875356354863976, −4.72798956152135839357449177468, −3.37888016125922144295156942006, −2.56280035894798757612720632296, −1.36931825023420674561066789334, 0,
1.36931825023420674561066789334, 2.56280035894798757612720632296, 3.37888016125922144295156942006, 4.72798956152135839357449177468, 5.37707931082636875356354863976, 5.60673425928209812679721395124, 6.48700258312266337684566136984, 6.83201829495193969412646199372, 7.67483948332540496550750895946
