| L(s) = 1 | − 0.357·3-s + 3.31·5-s − 7-s − 2.87·9-s − 11-s − 13-s − 1.18·15-s + 2.87·17-s − 0.480·19-s + 0.357·21-s − 8.87·23-s + 6.00·25-s + 2.09·27-s + 2.06·29-s + 8.14·31-s + 0.357·33-s − 3.31·35-s + 9.94·37-s + 0.357·39-s + 6.95·41-s − 5.10·43-s − 9.52·45-s − 3.30·47-s + 49-s − 1.02·51-s − 6.64·53-s − 3.31·55-s + ⋯ |
| L(s) = 1 | − 0.206·3-s + 1.48·5-s − 0.377·7-s − 0.957·9-s − 0.301·11-s − 0.277·13-s − 0.306·15-s + 0.697·17-s − 0.110·19-s + 0.0779·21-s − 1.85·23-s + 1.20·25-s + 0.403·27-s + 0.384·29-s + 1.46·31-s + 0.0621·33-s − 0.560·35-s + 1.63·37-s + 0.0572·39-s + 1.08·41-s − 0.778·43-s − 1.42·45-s − 0.482·47-s + 0.142·49-s − 0.143·51-s − 0.913·53-s − 0.447·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.969169711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.969169711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.357T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 + 0.480T + 19T^{2} \) |
| 23 | \( 1 + 8.87T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 - 8.14T + 31T^{2} \) |
| 37 | \( 1 - 9.94T + 37T^{2} \) |
| 41 | \( 1 - 6.95T + 41T^{2} \) |
| 43 | \( 1 + 5.10T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 1.62T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 9.23T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.271648773764873993523927622503, −7.993674227676676484747920858748, −6.60447624713091600430735604047, −6.23152888800793883037401518776, −5.57732668936503800964457467420, −4.95655213585286465519770510243, −3.79040318973845114062355722146, −2.66588958884368118945914929282, −2.19628759009659513985509412873, −0.798977643183734560082492824357,
0.798977643183734560082492824357, 2.19628759009659513985509412873, 2.66588958884368118945914929282, 3.79040318973845114062355722146, 4.95655213585286465519770510243, 5.57732668936503800964457467420, 6.23152888800793883037401518776, 6.60447624713091600430735604047, 7.993674227676676484747920858748, 8.271648773764873993523927622503
