| L(s) = 1 | + 3-s − 2·5-s + 9-s − 2·13-s − 2·15-s + 2·17-s − 19-s − 25-s + 27-s − 2·29-s + 4·31-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s − 2·45-s − 7·49-s + 2·51-s − 10·53-s − 57-s − 4·59-s + 2·61-s + 4·65-s − 12·67-s − 6·73-s − 75-s + 4·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 0.485·17-s − 0.229·19-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s − 49-s + 0.280·51-s − 1.37·53-s − 0.132·57-s − 0.520·59-s + 0.256·61-s + 0.496·65-s − 1.46·67-s − 0.702·73-s − 0.115·75-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.990283855407605113751096428643, −7.68112268536493640643463612492, −6.85794126628927735474787629285, −5.98961361242681266882536923090, −4.94287700126523316079777314529, −4.24642656992658091101795428526, −3.43913546512073985205242958174, −2.66863335230225144771302693759, −1.49462952262318723135222749012, 0,
1.49462952262318723135222749012, 2.66863335230225144771302693759, 3.43913546512073985205242958174, 4.24642656992658091101795428526, 4.94287700126523316079777314529, 5.98961361242681266882536923090, 6.85794126628927735474787629285, 7.68112268536493640643463612492, 7.990283855407605113751096428643
