| L(s) = 1 | − 5-s − 4·7-s − 3·9-s − 4·11-s + 2·13-s + 2·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s − 8·31-s + 4·35-s − 6·37-s − 6·41-s + 8·43-s + 3·45-s + 4·47-s + 9·49-s − 6·53-s + 4·55-s + 4·59-s + 2·61-s + 12·63-s − 2·65-s − 8·67-s − 6·73-s + 16·77-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s + 1.21·43-s + 0.447·45-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.248·65-s − 0.977·67-s − 0.702·73-s + 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−11.01108015186625578670651780817, −10.39655063289957855259191225662, −9.206637608661514184739644068270, −8.421702762202697929017375581976, −7.28963827504896405934712065825, −6.22671236180701133336548488807, −5.28966562607622137666656232279, −3.64413717916439174649981742998, −2.75667504176180688667939525456, 0,
2.75667504176180688667939525456, 3.64413717916439174649981742998, 5.28966562607622137666656232279, 6.22671236180701133336548488807, 7.28963827504896405934712065825, 8.421702762202697929017375581976, 9.206637608661514184739644068270, 10.39655063289957855259191225662, 11.01108015186625578670651780817
