| L(s) = 1 | − 67.2·5-s − 49·7-s + 17.2·11-s + 807.·13-s − 777.·17-s + 262.·19-s + 1.95e3·23-s + 1.39e3·25-s + 1.51e3·29-s + 5.47e3·31-s + 3.29e3·35-s + 6.58e3·37-s − 2.93e3·41-s − 975.·43-s − 2.29e4·47-s + 2.40e3·49-s − 2.64e3·53-s − 1.15e3·55-s − 2.25e4·59-s + 9.53e3·61-s − 5.42e4·65-s − 6.78e4·67-s + 1.01e4·71-s + 6.01e4·73-s − 842.·77-s − 2.77e4·79-s + 1.34e4·83-s + ⋯ |
| L(s) = 1 | − 1.20·5-s − 0.377·7-s + 0.0428·11-s + 1.32·13-s − 0.652·17-s + 0.166·19-s + 0.768·23-s + 0.445·25-s + 0.335·29-s + 1.02·31-s + 0.454·35-s + 0.791·37-s − 0.272·41-s − 0.0804·43-s − 1.51·47-s + 0.142·49-s − 0.129·53-s − 0.0515·55-s − 0.841·59-s + 0.328·61-s − 1.59·65-s − 1.84·67-s + 0.239·71-s + 1.32·73-s − 0.0162·77-s − 0.500·79-s + 0.214·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 5 | \( 1 + 67.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 17.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 807.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 777.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 262.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.95e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.58e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.93e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 975.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.64e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.53e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.86e4T + 8.58e9T^{2} \) |
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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
−9.629314795905935918413780618966, −8.621178129407207304023885656272, −8.002332842402755477921973223396, −6.92472577299283338810298725808, −6.12087543698603210981862035714, −4.72789175974615907954250042752, −3.83068470321456031051844240210, −2.92364711999085192770274088194, −1.20356547762687194784091125185, 0,
1.20356547762687194784091125185, 2.92364711999085192770274088194, 3.83068470321456031051844240210, 4.72789175974615907954250042752, 6.12087543698603210981862035714, 6.92472577299283338810298725808, 8.002332842402755477921973223396, 8.621178129407207304023885656272, 9.629314795905935918413780618966
