| L(s) = 1 | + 3·3-s + 5-s + 7-s + 6·9-s − 2·11-s + 3·15-s − 3·17-s + 6·19-s + 3·21-s + 4·23-s − 4·25-s + 9·27-s + 2·29-s + 4·31-s − 6·33-s + 35-s − 3·37-s + 5·43-s + 6·45-s + 13·47-s − 6·49-s − 9·51-s + 12·53-s − 2·55-s + 18·57-s − 10·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 0.603·11-s + 0.774·15-s − 0.727·17-s + 1.37·19-s + 0.654·21-s + 0.834·23-s − 4/5·25-s + 1.73·27-s + 0.371·29-s + 0.718·31-s − 1.04·33-s + 0.169·35-s − 0.493·37-s + 0.762·43-s + 0.894·45-s + 1.89·47-s − 6/7·49-s − 1.26·51-s + 1.64·53-s − 0.269·55-s + 2.38·57-s − 1.30·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.002843516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.002843516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.968437800976028493256072760850, −8.051660281056901581015263821725, −7.58433170795290376453101971689, −6.83942202332506672968213427091, −5.67644357149883273045642983440, −4.77110978952904042995749180644, −3.89239162271605294968540111677, −2.91875359123950367684627079786, −2.35658502852942708851276680816, −1.29336546110615612462449119400,
1.29336546110615612462449119400, 2.35658502852942708851276680816, 2.91875359123950367684627079786, 3.89239162271605294968540111677, 4.77110978952904042995749180644, 5.67644357149883273045642983440, 6.83942202332506672968213427091, 7.58433170795290376453101971689, 8.051660281056901581015263821725, 8.968437800976028493256072760850
