| L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 9-s + 6·13-s − 4·15-s − 4·17-s − 2·19-s + 2·21-s − 25-s + 4·27-s + 10·31-s − 2·35-s − 2·37-s − 12·39-s − 12·43-s + 2·45-s − 6·47-s + 49-s + 8·51-s + 6·53-s + 4·57-s − 4·59-s − 4·61-s − 63-s + 12·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 1.03·15-s − 0.970·17-s − 0.458·19-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 1.79·31-s − 0.338·35-s − 0.328·37-s − 1.92·39-s − 1.82·43-s + 0.298·45-s − 0.875·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.512·61-s − 0.125·63-s + 1.48·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.08384450148809, −14.00948586824951, −13.81704798165432, −13.21913494782329, −12.95596866904372, −12.19635868632897, −11.55660787873176, −11.38693917409908, −10.63658513815121, −10.30165303193217, −9.833999072066445, −9.053371345478145, −8.583199087531327, −8.168754265379330, −7.153582302861251, −6.483055916421887, −6.169215607803521, −6.024653513086353, −5.056462638713431, −4.747433874750181, −3.873325527519389, −3.235646843158298, −2.385248488072640, −1.645418373619752, −0.9144165330146104, 0,
0.9144165330146104, 1.645418373619752, 2.385248488072640, 3.235646843158298, 3.873325527519389, 4.747433874750181, 5.056462638713431, 6.024653513086353, 6.169215607803521, 6.483055916421887, 7.153582302861251, 8.168754265379330, 8.583199087531327, 9.053371345478145, 9.833999072066445, 10.30165303193217, 10.63658513815121, 11.38693917409908, 11.55660787873176, 12.19635868632897, 12.95596866904372, 13.21913494782329, 13.81704798165432, 14.00948586824951, 15.08384450148809
