| L(s) = 1 | − 5-s − 2·7-s − 2·13-s + 2·17-s − 4·19-s + 23-s + 25-s + 8·31-s + 2·35-s + 4·37-s + 4·41-s + 6·43-s − 3·49-s + 2·53-s + 6·59-s − 6·61-s + 2·65-s + 10·67-s − 6·71-s − 14·73-s + 4·79-s − 4·83-s − 2·85-s − 14·89-s + 4·91-s + 4·95-s + 16·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.43·31-s + 0.338·35-s + 0.657·37-s + 0.624·41-s + 0.914·43-s − 3/7·49-s + 0.274·53-s + 0.781·59-s − 0.768·61-s + 0.248·65-s + 1.22·67-s − 0.712·71-s − 1.63·73-s + 0.450·79-s − 0.439·83-s − 0.216·85-s − 1.48·89-s + 0.419·91-s + 0.410·95-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.50142682921084004308251223917, −6.72853355138153476167601282867, −6.20159596755314423306397362662, −5.40641505962952213769491325026, −4.50825471906334578236665309053, −3.97194064217666252160273578765, −2.99962132327439516733745905969, −2.44153142495080491261207041319, −1.10729857217634289120488027758, 0,
1.10729857217634289120488027758, 2.44153142495080491261207041319, 2.99962132327439516733745905969, 3.97194064217666252160273578765, 4.50825471906334578236665309053, 5.40641505962952213769491325026, 6.20159596755314423306397362662, 6.72853355138153476167601282867, 7.50142682921084004308251223917
