| L(s) = 1 | + 2.62·3-s − 1.67·5-s + 2.55·7-s + 3.87·9-s + 1.82·11-s − 0.173·13-s − 4.40·15-s − 1.35·17-s + 2.87·19-s + 6.70·21-s + 6.29·23-s − 2.17·25-s + 2.30·27-s − 6.08·29-s + 8.59·31-s + 4.79·33-s − 4.29·35-s − 1.42·37-s − 0.454·39-s + 8.16·41-s − 43-s − 6.51·45-s + 9.17·47-s − 0.461·49-s − 3.54·51-s + 3.53·53-s − 3.06·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s − 0.751·5-s + 0.966·7-s + 1.29·9-s + 0.550·11-s − 0.0480·13-s − 1.13·15-s − 0.328·17-s + 0.660·19-s + 1.46·21-s + 1.31·23-s − 0.435·25-s + 0.443·27-s − 1.12·29-s + 1.54·31-s + 0.833·33-s − 0.725·35-s − 0.234·37-s − 0.0728·39-s + 1.27·41-s − 0.152·43-s − 0.970·45-s + 1.33·47-s − 0.0658·49-s − 0.496·51-s + 0.485·53-s − 0.413·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.319982914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.319982914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 + 1.67T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 + 0.173T + 13T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 - 2.87T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 - 8.59T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 - 8.16T + 41T^{2} \) |
| 47 | \( 1 - 9.17T + 47T^{2} \) |
| 53 | \( 1 - 3.53T + 53T^{2} \) |
| 59 | \( 1 - 8.46T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 6.44T + 73T^{2} \) |
| 79 | \( 1 + 9.87T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−8.774639935477343951831462771874, −8.067706645167156290900292684779, −7.55262227250290098404696852027, −6.93468421465525176826379530401, −5.64744599063421853941865923506, −4.57449310625770715456911441823, −3.97790021268223105384546081087, −3.12933318648445165309983314961, −2.24871852263045313911271327686, −1.15089616231130350260010963856,
1.15089616231130350260010963856, 2.24871852263045313911271327686, 3.12933318648445165309983314961, 3.97790021268223105384546081087, 4.57449310625770715456911441823, 5.64744599063421853941865923506, 6.93468421465525176826379530401, 7.55262227250290098404696852027, 8.067706645167156290900292684779, 8.774639935477343951831462771874
