| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 6·11-s + 12-s − 4·13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s + 21-s + 6·22-s − 23-s − 24-s + 25-s + 4·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 1.27·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.78844437531077, −13.89008958876934, −13.44332517720600, −13.00667223176805, −12.52539764533131, −12.08210825574281, −11.26532070112769, −10.74722534290345, −10.48960480943400, −9.956664215173367, −9.374525705072295, −8.991689130419675, −8.410541121729538, −7.842680761400633, −7.456657927453526, −7.144755999421428, −6.269149202595982, −5.668203716424800, −5.169694767761840, −4.608288234659486, −3.892970429262524, −2.938586461622653, −2.652532499514128, −1.899979904608380, −1.599222173373442, 0, 0,
1.599222173373442, 1.899979904608380, 2.652532499514128, 2.938586461622653, 3.892970429262524, 4.608288234659486, 5.169694767761840, 5.668203716424800, 6.269149202595982, 7.144755999421428, 7.456657927453526, 7.842680761400633, 8.410541121729538, 8.991689130419675, 9.374525705072295, 9.956664215173367, 10.48960480943400, 10.74722534290345, 11.26532070112769, 12.08210825574281, 12.52539764533131, 13.00667223176805, 13.44332517720600, 13.89008958876934, 14.78844437531077
