L(s) = 1 | − 2.76·2-s − 0.148·3-s + 5.62·4-s + 2.53·5-s + 0.409·6-s − 9.99·8-s − 2.97·9-s − 6.99·10-s + 1.58·11-s − 0.834·12-s − 6.17·13-s − 0.376·15-s + 16.3·16-s + 3.03·17-s + 8.22·18-s + 19-s + 14.2·20-s − 4.37·22-s + 6.07·23-s + 1.48·24-s + 1.42·25-s + 17.0·26-s + 0.887·27-s + 1.46·29-s + 1.03·30-s + 7.46·31-s − 25.1·32-s + ⋯ |
L(s) = 1 | − 1.95·2-s − 0.0856·3-s + 2.81·4-s + 1.13·5-s + 0.167·6-s − 3.53·8-s − 0.992·9-s − 2.21·10-s + 0.477·11-s − 0.240·12-s − 1.71·13-s − 0.0971·15-s + 4.08·16-s + 0.735·17-s + 1.93·18-s + 0.229·19-s + 3.18·20-s − 0.931·22-s + 1.26·23-s + 0.302·24-s + 0.284·25-s + 3.34·26-s + 0.170·27-s + 0.271·29-s + 0.189·30-s + 1.34·31-s − 4.44·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7082210590\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7082210590\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 + 0.148T + 3T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 + 6.17T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 23 | \( 1 - 6.07T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 37 | \( 1 + 0.493T + 37T^{2} \) |
| 41 | \( 1 - 6.46T + 41T^{2} \) |
| 43 | \( 1 - 0.636T + 43T^{2} \) |
| 47 | \( 1 + 3.56T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 3.53T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 + 0.975T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 0.284T + 83T^{2} \) |
| 89 | \( 1 - 6.16T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−9.995539737082836632616463530511, −9.260151178948537476886790401445, −8.670535336027581357659552303672, −7.67581709462592597920376516498, −6.92122927177060653780411062275, −6.06289254305189573831902900072, −5.24721000322024092892400761537, −2.92621925691212617368835394855, −2.26197585650782196654909701567, −0.886430507002185381784286522537,
0.886430507002185381784286522537, 2.26197585650782196654909701567, 2.92621925691212617368835394855, 5.24721000322024092892400761537, 6.06289254305189573831902900072, 6.92122927177060653780411062275, 7.67581709462592597920376516498, 8.670535336027581357659552303672, 9.260151178948537476886790401445, 9.995539737082836632616463530511