| L(s) = 1 | − 4·2-s + 12·3-s + 16·4-s − 25·5-s − 48·6-s + 54·7-s − 64·8-s − 99·9-s + 100·10-s − 121·11-s + 192·12-s − 540·13-s − 216·14-s − 300·15-s + 256·16-s + 340·17-s + 396·18-s − 952·19-s − 400·20-s + 648·21-s + 484·22-s + 1.09e3·23-s − 768·24-s + 625·25-s + 2.16e3·26-s − 4.10e3·27-s + 864·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.769·3-s + 1/2·4-s − 0.447·5-s − 0.544·6-s + 0.416·7-s − 0.353·8-s − 0.407·9-s + 0.316·10-s − 0.301·11-s + 0.384·12-s − 0.886·13-s − 0.294·14-s − 0.344·15-s + 1/4·16-s + 0.285·17-s + 0.288·18-s − 0.604·19-s − 0.223·20-s + 0.320·21-s + 0.213·22-s + 0.430·23-s − 0.272·24-s + 1/5·25-s + 0.626·26-s − 1.08·27-s + 0.208·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| 11 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 4 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 54 T + p^{5} T^{2} \) |
| 13 | \( 1 + 540 T + p^{5} T^{2} \) |
| 17 | \( 1 - 20 p T + p^{5} T^{2} \) |
| 19 | \( 1 + 952 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1092 T + p^{5} T^{2} \) |
| 29 | \( 1 + 62 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7560 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9186 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6818 T + p^{5} T^{2} \) |
| 43 | \( 1 + 13310 T + p^{5} T^{2} \) |
| 47 | \( 1 + 22420 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19654 T + p^{5} T^{2} \) |
| 59 | \( 1 - 48292 T + p^{5} T^{2} \) |
| 61 | \( 1 - 17530 T + p^{5} T^{2} \) |
| 67 | \( 1 + 35344 T + p^{5} T^{2} \) |
| 71 | \( 1 + 22912 T + p^{5} T^{2} \) |
| 73 | \( 1 - 47852 T + p^{5} T^{2} \) |
| 79 | \( 1 - 52396 T + p^{5} T^{2} \) |
| 83 | \( 1 - 7890 T + p^{5} T^{2} \) |
| 89 | \( 1 - 41958 T + p^{5} T^{2} \) |
| 97 | \( 1 + 37602 T + p^{5} T^{2} \) |
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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
−12.02661667512864949047564501864, −11.06143890428613164497064308525, −9.901003029529095983874824832745, −8.755048912485438801279000836892, −8.027534617609041363111495345881, −6.98729395990191357659298751840, −5.20265702733817596526387407457, −3.39750248986186982393695630272, −2.03377158952164860109626503500, 0,
2.03377158952164860109626503500, 3.39750248986186982393695630272, 5.20265702733817596526387407457, 6.98729395990191357659298751840, 8.027534617609041363111495345881, 8.755048912485438801279000836892, 9.901003029529095983874824832745, 11.06143890428613164497064308525, 12.02661667512864949047564501864
