| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 2·13-s + 15-s + 17-s + 4·19-s − 4·23-s + 25-s − 27-s + 2·29-s + 4·31-s + 4·33-s − 6·37-s − 2·39-s − 10·41-s + 8·43-s − 45-s − 7·49-s − 51-s + 6·53-s + 4·55-s − 4·57-s + 8·59-s + 10·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.696·33-s − 0.986·37-s − 0.320·39-s − 1.56·41-s + 1.21·43-s − 0.149·45-s − 49-s − 0.140·51-s + 0.824·53-s + 0.539·55-s − 0.529·57-s + 1.04·59-s + 1.28·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.156824574293896679574878503197, −7.28008628923442455115953124728, −6.67853486026176323485637173922, −5.64165192346952587210203852330, −5.25363351909920656851833527800, −4.29642718133962735402779532430, −3.45840633167579953368299139748, −2.52209682764452214813693572518, −1.23484859129809693887858355508, 0,
1.23484859129809693887858355508, 2.52209682764452214813693572518, 3.45840633167579953368299139748, 4.29642718133962735402779532430, 5.25363351909920656851833527800, 5.64165192346952587210203852330, 6.67853486026176323485637173922, 7.28008628923442455115953124728, 8.156824574293896679574878503197
