| L(s) = 1 | + 0.0954·3-s + 0.581·5-s − 2.99·9-s − 4.59·11-s + 5.44·13-s + 0.0554·15-s + 0.983·17-s − 0.599·19-s + 5.72·23-s − 4.66·25-s − 0.571·27-s + 6.69·29-s − 1.57·31-s − 0.438·33-s − 8.89·37-s + 0.519·39-s + 41-s − 5.24·43-s − 1.73·45-s + 7.66·47-s + 0.0938·51-s − 11.3·53-s − 2.66·55-s − 0.0572·57-s − 7.53·59-s − 5.90·61-s + 3.16·65-s + ⋯ |
| L(s) = 1 | + 0.0550·3-s + 0.259·5-s − 0.996·9-s − 1.38·11-s + 1.51·13-s + 0.0143·15-s + 0.238·17-s − 0.137·19-s + 1.19·23-s − 0.932·25-s − 0.110·27-s + 1.24·29-s − 0.283·31-s − 0.0762·33-s − 1.46·37-s + 0.0832·39-s + 0.156·41-s − 0.799·43-s − 0.259·45-s + 1.11·47-s + 0.0131·51-s − 1.55·53-s − 0.359·55-s − 0.00758·57-s − 0.980·59-s − 0.755·61-s + 0.392·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0954T + 3T^{2} \) |
| 5 | \( 1 - 0.581T + 5T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 - 0.983T + 17T^{2} \) |
| 19 | \( 1 + 0.599T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 - 6.69T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 - 7.66T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 7.53T + 59T^{2} \) |
| 61 | \( 1 + 5.90T + 61T^{2} \) |
| 67 | \( 1 - 2.45T + 67T^{2} \) |
| 71 | \( 1 + 8.01T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 1.44T + 97T^{2} \) |
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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
−7.71331406748227483083194580591, −6.62265089740644249869804097745, −6.12169277354440820090389425069, −5.34596212236964460443986007505, −4.91617966797354879460222252366, −3.69949979739857448593293552642, −3.10269710744142992151629247460, −2.33968161346849064706877655819, −1.25735973614126427776282669824, 0,
1.25735973614126427776282669824, 2.33968161346849064706877655819, 3.10269710744142992151629247460, 3.69949979739857448593293552642, 4.91617966797354879460222252366, 5.34596212236964460443986007505, 6.12169277354440820090389425069, 6.62265089740644249869804097745, 7.71331406748227483083194580591
