L(s) = 1 | − 5-s − 4·11-s − 2·13-s − 2·17-s − 19-s + 4·23-s + 25-s + 6·29-s − 4·31-s + 6·37-s − 10·41-s − 4·43-s − 12·47-s − 7·49-s + 6·53-s + 4·55-s + 12·59-s + 2·61-s + 2·65-s + 4·67-s + 8·71-s − 6·73-s + 4·79-s + 12·83-s + 2·85-s − 10·89-s + 95-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s − 49-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.450·79-s + 1.31·83-s + 0.216·85-s − 1.05·89-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.70256732075949, −14.30910047057656, −13.42931427499342, −13.16427482824578, −12.75119460101451, −12.13687968974046, −11.51544033856593, −11.18056321347570, −10.58444741635190, −9.955150041980534, −9.743496419889676, −8.764710053330327, −8.437830101448229, −7.933562616240356, −7.323427181905562, −6.792484911037260, −6.326714861914369, −5.420994344500249, −4.925532731299856, −4.636323608129330, −3.670508019885303, −3.150666955757874, −2.476939306821640, −1.864226832490663, −0.7783375238680073, 0,
0.7783375238680073, 1.864226832490663, 2.476939306821640, 3.150666955757874, 3.670508019885303, 4.636323608129330, 4.925532731299856, 5.420994344500249, 6.326714861914369, 6.792484911037260, 7.323427181905562, 7.933562616240356, 8.437830101448229, 8.764710053330327, 9.743496419889676, 9.955150041980534, 10.58444741635190, 11.18056321347570, 11.51544033856593, 12.13687968974046, 12.75119460101451, 13.16427482824578, 13.42931427499342, 14.30910047057656, 14.70256732075949