| L(s) = 1 | + 3-s + 4·5-s + 2·7-s + 9-s − 4·11-s − 2·13-s + 4·15-s − 2·17-s − 8·19-s + 2·21-s + 4·23-s + 11·25-s + 27-s − 6·31-s − 4·33-s + 8·35-s + 2·37-s − 2·39-s + 6·41-s + 4·45-s + 4·47-s − 3·49-s − 2·51-s − 16·55-s − 8·57-s + 4·59-s − 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 1.03·15-s − 0.485·17-s − 1.83·19-s + 0.436·21-s + 0.834·23-s + 11/5·25-s + 0.192·27-s − 1.07·31-s − 0.696·33-s + 1.35·35-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.596·45-s + 0.583·47-s − 3/7·49-s − 0.280·51-s − 2.15·55-s − 1.05·57-s + 0.520·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−10.90639027040777849278794598785, −10.48664968251819049733732284340, −9.433707596930861266358550422356, −8.743300071949026671666068984095, −7.69025257463999263217129645276, −6.52923673370585434887828168703, −5.46781891072947205774781799219, −4.59332442534898507060628341827, −2.63009703571906191631419054606, −1.91249343134731656186986110897,
1.91249343134731656186986110897, 2.63009703571906191631419054606, 4.59332442534898507060628341827, 5.46781891072947205774781799219, 6.52923673370585434887828168703, 7.69025257463999263217129645276, 8.743300071949026671666068984095, 9.433707596930861266358550422356, 10.48664968251819049733732284340, 10.90639027040777849278794598785
