| L(s) = 1 | − 3.02·3-s − 1.69·5-s + 7-s + 6.12·9-s − 1.32·11-s − 1.69·13-s + 5.12·15-s + 2·17-s − 3.02·19-s − 3.02·21-s − 5.12·23-s − 2.12·25-s − 9.43·27-s + 6.04·29-s − 10.2·31-s + 4·33-s − 1.69·35-s + 6.04·37-s + 5.12·39-s − 4.24·41-s − 1.32·43-s − 10.3·45-s + 49-s − 6.04·51-s − 2.64·53-s + 2.24·55-s + 9.12·57-s + ⋯ |
| L(s) = 1 | − 1.74·3-s − 0.758·5-s + 0.377·7-s + 2.04·9-s − 0.399·11-s − 0.470·13-s + 1.32·15-s + 0.485·17-s − 0.692·19-s − 0.659·21-s − 1.06·23-s − 0.424·25-s − 1.81·27-s + 1.12·29-s − 1.84·31-s + 0.696·33-s − 0.286·35-s + 0.993·37-s + 0.820·39-s − 0.663·41-s − 0.201·43-s − 1.54·45-s + 0.142·49-s − 0.845·51-s − 0.363·53-s + 0.302·55-s + 1.20·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5156663918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5156663918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 3.02T + 3T^{2} \) |
| 5 | \( 1 + 1.69T + 5T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 - 6.04T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 6.04T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.64T + 53T^{2} \) |
| 59 | \( 1 + 0.371T + 59T^{2} \) |
| 61 | \( 1 + 1.69T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 5.66T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−9.537406211227885560107521001503, −8.220085393638829175866279820567, −7.63640993049787425895866351978, −6.78598577855935644223605254524, −5.99583877030064121998982458951, −5.22811483155944353768458301191, −4.53888768328166144068059511762, −3.68422365984952772772831428934, −1.99062849331431991659766749569, −0.52399927878331299383355506105,
0.52399927878331299383355506105, 1.99062849331431991659766749569, 3.68422365984952772772831428934, 4.53888768328166144068059511762, 5.22811483155944353768458301191, 5.99583877030064121998982458951, 6.78598577855935644223605254524, 7.63640993049787425895866351978, 8.220085393638829175866279820567, 9.537406211227885560107521001503
