L(s) = 1 | − 2-s + 0.519·3-s + 4-s − 3.34·5-s − 0.519·6-s + 4.82·7-s − 8-s − 2.73·9-s + 3.34·10-s − 2.93·11-s + 0.519·12-s − 1.57·13-s − 4.82·14-s − 1.73·15-s + 16-s − 0.289·17-s + 2.73·18-s − 5.67·19-s − 3.34·20-s + 2.50·21-s + 2.93·22-s − 5.60·23-s − 0.519·24-s + 6.17·25-s + 1.57·26-s − 2.97·27-s + 4.82·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.299·3-s + 0.5·4-s − 1.49·5-s − 0.212·6-s + 1.82·7-s − 0.353·8-s − 0.910·9-s + 1.05·10-s − 0.885·11-s + 0.149·12-s − 0.438·13-s − 1.29·14-s − 0.448·15-s + 0.250·16-s − 0.0702·17-s + 0.643·18-s − 1.30·19-s − 0.747·20-s + 0.547·21-s + 0.626·22-s − 1.16·23-s − 0.106·24-s + 1.23·25-s + 0.309·26-s − 0.572·27-s + 0.912·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6568931043\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6568931043\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
good | 3 | \( 1 - 0.519T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 + 0.289T + 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 + 5.60T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 - 4.63T + 41T^{2} \) |
| 43 | \( 1 - 4.04T + 43T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 + 4.53T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 5.24T + 89T^{2} \) |
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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
−8.159351476808599193780779841399, −7.65335198513199684511428653296, −7.15430916313375950871338102722, −5.87351551675733651366120611353, −5.20219649407308079300957927826, −4.31932895451575441199061435447, −3.73136090136245729315469083349, −2.49693157106835147236927387049, −1.95086341533898376051773506370, −0.44381806081945235703297228651,
0.44381806081945235703297228651, 1.95086341533898376051773506370, 2.49693157106835147236927387049, 3.73136090136245729315469083349, 4.31932895451575441199061435447, 5.20219649407308079300957927826, 5.87351551675733651366120611353, 7.15430916313375950871338102722, 7.65335198513199684511428653296, 8.159351476808599193780779841399