| L(s) = 1 | − 3-s + 9-s + 4·11-s − 6·13-s + 6·17-s + 4·23-s − 27-s − 6·29-s − 4·33-s − 2·37-s + 6·39-s − 2·41-s − 4·43-s + 4·47-s − 6·51-s + 6·53-s − 12·59-s + 10·61-s − 4·67-s − 4·69-s + 8·71-s − 14·73-s + 8·79-s + 81-s − 12·83-s + 6·87-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 0.840·51-s + 0.824·53-s − 1.56·59-s + 1.28·61-s − 0.488·67-s − 0.481·69-s + 0.949·71-s − 1.63·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s + 0.643·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.84124931413407, −13.33643881443686, −12.63586930027470, −12.30322747535821, −11.98506410671782, −11.48112180749293, −11.01910416391183, −10.36835763291234, −9.817034528871117, −9.649929668187192, −8.985679200999273, −8.495824678939200, −7.671393337662023, −7.300342975793330, −6.977678688303317, −6.297545653629903, −5.702019388353719, −5.263643679848898, −4.761702399874075, −4.150343059385815, −3.519620526506032, −2.990279345747777, −2.166276507020384, −1.502704072760845, −0.8560391465956767, 0,
0.8560391465956767, 1.502704072760845, 2.166276507020384, 2.990279345747777, 3.519620526506032, 4.150343059385815, 4.761702399874075, 5.263643679848898, 5.702019388353719, 6.297545653629903, 6.977678688303317, 7.300342975793330, 7.671393337662023, 8.495824678939200, 8.985679200999273, 9.649929668187192, 9.817034528871117, 10.36835763291234, 11.01910416391183, 11.48112180749293, 11.98506410671782, 12.30322747535821, 12.63586930027470, 13.33643881443686, 13.84124931413407
