L(s) = 1 | − 2·4-s − 5-s + 7-s − 4·13-s + 4·16-s − 17-s + 6·19-s + 2·20-s − 6·23-s − 4·25-s − 2·28-s − 29-s + 4·31-s − 35-s − 6·37-s − 6·41-s − 5·43-s − 7·47-s + 49-s + 8·52-s + 4·53-s + 11·59-s − 8·61-s − 8·64-s + 4·65-s + 67-s + 2·68-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s + 0.377·7-s − 1.10·13-s + 16-s − 0.242·17-s + 1.37·19-s + 0.447·20-s − 1.25·23-s − 4/5·25-s − 0.377·28-s − 0.185·29-s + 0.718·31-s − 0.169·35-s − 0.986·37-s − 0.937·41-s − 0.762·43-s − 1.02·47-s + 1/7·49-s + 1.10·52-s + 0.549·53-s + 1.43·59-s − 1.02·61-s − 64-s + 0.496·65-s + 0.122·67-s + 0.242·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 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
−13.57118140100875, −12.98088648705919, −12.42431653527021, −12.03300553272450, −11.61090150227449, −11.37751550597566, −10.31102027923966, −10.18986551561740, −9.696835838140957, −9.328255841553062, −8.610664494809162, −8.215670830866094, −7.921029884555809, −7.274591198398818, −6.965227039335769, −6.145168905543435, −5.547068512841897, −5.162320630610332, −4.718153973572575, −4.161430801171776, −3.681793604482147, −3.159378281812428, −2.455550769471187, −1.708852512053143, −1.126749176102283, 0, 0,
1.126749176102283, 1.708852512053143, 2.455550769471187, 3.159378281812428, 3.681793604482147, 4.161430801171776, 4.718153973572575, 5.162320630610332, 5.547068512841897, 6.145168905543435, 6.965227039335769, 7.274591198398818, 7.921029884555809, 8.215670830866094, 8.610664494809162, 9.328255841553062, 9.696835838140957, 10.18986551561740, 10.31102027923966, 11.37751550597566, 11.61090150227449, 12.03300553272450, 12.42431653527021, 12.98088648705919, 13.57118140100875