| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s − 11-s + 2·13-s − 2·15-s − 4·17-s − 6·19-s + 2·21-s − 25-s − 27-s + 8·29-s + 33-s − 4·35-s − 10·37-s − 2·39-s + 8·41-s + 2·43-s + 2·45-s − 8·47-s − 3·49-s + 4·51-s + 2·53-s − 2·55-s + 6·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.516·15-s − 0.970·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.174·33-s − 0.676·35-s − 1.64·37-s − 0.320·39-s + 1.24·41-s + 0.304·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s − 0.269·55-s + 0.794·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126852 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.58760955879774, −13.20612221575989, −12.89747814583004, −12.35893228952460, −11.93873383427208, −11.09837253327688, −10.93101642135562, −10.32013497363073, −9.976257892195560, −9.464813950278765, −8.905757586323797, −8.416795465839633, −7.981819704406012, −6.970340636155780, −6.713587744547758, −6.325946770537594, −5.801191687013291, −5.339740204490234, −4.615251568740218, −4.201634964226513, −3.489796757220532, −2.792059785695254, −2.165431026587630, −1.679891094468205, −0.7409721318191995, 0,
0.7409721318191995, 1.679891094468205, 2.165431026587630, 2.792059785695254, 3.489796757220532, 4.201634964226513, 4.615251568740218, 5.339740204490234, 5.801191687013291, 6.325946770537594, 6.713587744547758, 6.970340636155780, 7.981819704406012, 8.416795465839633, 8.905757586323797, 9.464813950278765, 9.976257892195560, 10.32013497363073, 10.93101642135562, 11.09837253327688, 11.93873383427208, 12.35893228952460, 12.89747814583004, 13.20612221575989, 13.58760955879774
