| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 2·13-s + 15-s + 6·19-s + 4·21-s − 2·23-s + 25-s − 27-s − 10·29-s + 8·31-s − 4·33-s + 4·35-s − 6·37-s + 2·39-s + 2·41-s − 45-s + 10·47-s + 9·49-s + 6·53-s − 4·55-s − 6·57-s − 4·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s + 1.37·19-s + 0.872·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.696·33-s + 0.676·35-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 0.149·45-s + 1.45·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 0.794·57-s − 0.512·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.992883565045267623959574889367, −7.14171916920406481852093221362, −6.74376258764267807585390205531, −5.92663151211549582068628632033, −5.25940207031282311952344051521, −4.08213967612376426073032732169, −3.61521951901471238783945328372, −2.62673584549065035587996041061, −1.17018441156581271676063282459, 0,
1.17018441156581271676063282459, 2.62673584549065035587996041061, 3.61521951901471238783945328372, 4.08213967612376426073032732169, 5.25940207031282311952344051521, 5.92663151211549582068628632033, 6.74376258764267807585390205531, 7.14171916920406481852093221362, 7.992883565045267623959574889367
