| L(s) = 1 | + 3-s + 4·7-s + 9-s − 4·11-s − 4·13-s + 4·21-s − 4·23-s + 27-s − 6·29-s + 4·31-s − 4·33-s + 2·37-s − 4·39-s + 2·41-s + 43-s − 8·47-s + 9·49-s + 6·53-s + 4·59-s − 4·61-s + 4·63-s + 8·67-s − 4·69-s − 4·71-s + 10·73-s − 16·77-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.872·21-s − 0.834·23-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.640·39-s + 0.312·41-s + 0.152·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s − 0.512·61-s + 0.503·63-s + 0.977·67-s − 0.481·69-s − 0.474·71-s + 1.17·73-s − 1.82·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.670020576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.670020576\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.44905261019418, −14.09142303538956, −13.56577639751325, −12.93810897941474, −12.56561956235535, −11.79881876430775, −11.50935065284133, −10.82241657827307, −10.37473850650101, −9.804899817559059, −9.352091154118724, −8.530745306673847, −8.150708953547150, −7.700524656408863, −7.389654910549342, −6.620830545546995, −5.738605228271687, −5.229641753792626, −4.757718065435974, −4.232932002179435, −3.468154989752984, −2.580799795458357, −2.213117929021005, −1.586645073416971, −0.5298913387586935,
0.5298913387586935, 1.586645073416971, 2.213117929021005, 2.580799795458357, 3.468154989752984, 4.232932002179435, 4.757718065435974, 5.229641753792626, 5.738605228271687, 6.620830545546995, 7.389654910549342, 7.700524656408863, 8.150708953547150, 8.530745306673847, 9.352091154118724, 9.804899817559059, 10.37473850650101, 10.82241657827307, 11.50935065284133, 11.79881876430775, 12.56561956235535, 12.93810897941474, 13.56577639751325, 14.09142303538956, 14.44905261019418
