| L(s) = 1 | − 2-s − 4-s + 2·5-s − 2·7-s + 3·8-s − 2·10-s − 11-s − 13-s + 2·14-s − 16-s − 4·17-s + 6·19-s − 2·20-s + 22-s − 8·23-s − 25-s + 26-s + 2·28-s + 10·31-s − 5·32-s + 4·34-s − 4·35-s − 6·37-s − 6·38-s + 6·40-s + 2·41-s + 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.755·7-s + 1.06·8-s − 0.632·10-s − 0.301·11-s − 0.277·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.447·20-s + 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.196·26-s + 0.377·28-s + 1.79·31-s − 0.883·32-s + 0.685·34-s − 0.676·35-s − 0.986·37-s − 0.973·38-s + 0.948·40-s + 0.312·41-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.532176920753137890987584145039, −8.543594329488774875715276800278, −7.83220216080835191171414546631, −6.82953781328202297194246907745, −5.97434372200033290838479265228, −5.07870141538241040440322366560, −4.08615501466508972346151437796, −2.80419701165731063280055544590, −1.59146148473310073061258647208, 0,
1.59146148473310073061258647208, 2.80419701165731063280055544590, 4.08615501466508972346151437796, 5.07870141538241040440322366560, 5.97434372200033290838479265228, 6.82953781328202297194246907745, 7.83220216080835191171414546631, 8.543594329488774875715276800278, 9.532176920753137890987584145039
