| L(s) = 1 | + 2-s + 3-s + 4-s + 0.732·5-s + 6-s − 7-s + 8-s + 9-s + 0.732·10-s + 1.73·11-s + 12-s − 14-s + 0.732·15-s + 16-s + 3.73·17-s + 18-s + 19-s + 0.732·20-s − 21-s + 1.73·22-s − 3.46·23-s + 24-s − 4.46·25-s + 27-s − 28-s + 6.46·29-s + 0.732·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.327·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.231·10-s + 0.522·11-s + 0.288·12-s − 0.267·14-s + 0.189·15-s + 0.250·16-s + 0.905·17-s + 0.235·18-s + 0.229·19-s + 0.163·20-s − 0.218·21-s + 0.369·22-s − 0.722·23-s + 0.204·24-s − 0.892·25-s + 0.192·27-s − 0.188·28-s + 1.20·29-s + 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.662240529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.662240529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 + 2.46T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 + 0.928T + 59T^{2} \) |
| 61 | \( 1 - 5.19T + 61T^{2} \) |
| 67 | \( 1 + 8.92T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 - 2.07T + 79T^{2} \) |
| 83 | \( 1 - 0.196T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.82656727792421263054484227059, −7.24678087820468346303546359809, −6.32424763165236096585908157832, −5.97103895807578313446061711528, −5.04763777981150501638593252012, −4.23079988810565550694857511535, −3.58039015171998361515897494690, −2.83652549533317190682432251823, −2.03331410962785072792703451352, −0.997456614711291038334733264077,
0.997456614711291038334733264077, 2.03331410962785072792703451352, 2.83652549533317190682432251823, 3.58039015171998361515897494690, 4.23079988810565550694857511535, 5.04763777981150501638593252012, 5.97103895807578313446061711528, 6.32424763165236096585908157832, 7.24678087820468346303546359809, 7.82656727792421263054484227059
