| L(s) = 1 | − 3-s − 3·5-s − 4·7-s − 2·9-s + 2·13-s + 3·15-s + 8·17-s + 6·19-s + 4·21-s − 5·23-s + 4·25-s + 5·27-s − 4·29-s − 31-s + 12·35-s + 3·37-s − 2·39-s + 6·41-s − 6·43-s + 6·45-s + 12·47-s + 9·49-s − 8·51-s − 6·53-s − 6·57-s − 3·59-s + 8·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 1.51·7-s − 2/3·9-s + 0.554·13-s + 0.774·15-s + 1.94·17-s + 1.37·19-s + 0.872·21-s − 1.04·23-s + 4/5·25-s + 0.962·27-s − 0.742·29-s − 0.179·31-s + 2.02·35-s + 0.493·37-s − 0.320·39-s + 0.937·41-s − 0.914·43-s + 0.894·45-s + 1.75·47-s + 9/7·49-s − 1.12·51-s − 0.824·53-s − 0.794·57-s − 0.390·59-s + 1.00·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.78122525591253568263543482390, −7.62722480655714859723348506405, −6.55308364199828309630932268801, −5.84688954059027242125847376557, −5.34340324995592983674267035014, −4.03192176008940410390660094975, −3.45638227499885283826740836493, −2.89719888783107338887618151319, −0.983028169290543283320284215782, 0,
0.983028169290543283320284215782, 2.89719888783107338887618151319, 3.45638227499885283826740836493, 4.03192176008940410390660094975, 5.34340324995592983674267035014, 5.84688954059027242125847376557, 6.55308364199828309630932268801, 7.62722480655714859723348506405, 7.78122525591253568263543482390
