| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 3·11-s − 5·13-s − 2·14-s + 16-s − 6·17-s − 4·19-s − 3·22-s + 3·23-s − 5·26-s − 2·28-s + 2·31-s + 32-s − 6·34-s − 11·37-s − 4·38-s + 6·41-s + 4·43-s − 3·44-s + 3·46-s − 3·47-s − 3·49-s − 5·52-s + 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.904·11-s − 1.38·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.639·22-s + 0.625·23-s − 0.980·26-s − 0.377·28-s + 0.359·31-s + 0.176·32-s − 1.02·34-s − 1.80·37-s − 0.648·38-s + 0.937·41-s + 0.609·43-s − 0.452·44-s + 0.442·46-s − 0.437·47-s − 3/7·49-s − 0.693·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 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 | \( 1 - 7 T + p T^{2} \) |
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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
−9.216241784045937440567684994224, −8.401851898487718919448316407101, −7.23314961345493011262006469397, −6.81439056322595129908795841841, −5.77545655198984264104456477800, −4.91136839351844552302763688534, −4.14304652178812049273867558691, −2.88562053703052468737324936302, −2.21669020630148497437590825476, 0,
2.21669020630148497437590825476, 2.88562053703052468737324936302, 4.14304652178812049273867558691, 4.91136839351844552302763688534, 5.77545655198984264104456477800, 6.81439056322595129908795841841, 7.23314961345493011262006469397, 8.401851898487718919448316407101, 9.216241784045937440567684994224
