| L(s) = 1 | + 2.48·2-s − 5.42·3-s − 1.80·4-s + 4.00·5-s − 13.5·6-s + 7·7-s − 24.4·8-s + 2.43·9-s + 9.96·10-s + 9.77·12-s + 51.9·13-s + 17.4·14-s − 21.7·15-s − 46.3·16-s − 59.1·17-s + 6.06·18-s − 13.7·19-s − 7.21·20-s − 37.9·21-s − 105.·23-s + 132.·24-s − 108.·25-s + 129.·26-s + 133.·27-s − 12.6·28-s − 106.·29-s − 54.0·30-s + ⋯ |
| L(s) = 1 | + 0.880·2-s − 1.04·3-s − 0.225·4-s + 0.358·5-s − 0.919·6-s + 0.377·7-s − 1.07·8-s + 0.0902·9-s + 0.315·10-s + 0.235·12-s + 1.10·13-s + 0.332·14-s − 0.373·15-s − 0.723·16-s − 0.843·17-s + 0.0794·18-s − 0.165·19-s − 0.0806·20-s − 0.394·21-s − 0.954·23-s + 1.12·24-s − 0.871·25-s + 0.974·26-s + 0.949·27-s − 0.0851·28-s − 0.684·29-s − 0.329·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.609322914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609322914\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.48T + 8T^{2} \) |
| 3 | \( 1 + 5.42T + 27T^{2} \) |
| 5 | \( 1 - 4.00T + 125T^{2} \) |
| 13 | \( 1 - 51.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 134.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 63.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 233.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 639.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 635.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 64.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 118.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 633.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 677.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 205.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 642.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 527.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 795.T + 9.12e5T^{2} \) |
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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
−9.959803487519015853948514122640, −8.919866431381794476886903221783, −8.211387972420303681502792767476, −6.75842096354881219637587277142, −5.93688156227225061180796605519, −5.54075121820139536138566025007, −4.49395680853310901576187267251, −3.73511353127276419596727485268, −2.24809501250796156592893204049, −0.63899910880212363937277447871,
0.63899910880212363937277447871, 2.24809501250796156592893204049, 3.73511353127276419596727485268, 4.49395680853310901576187267251, 5.54075121820139536138566025007, 5.93688156227225061180796605519, 6.75842096354881219637587277142, 8.211387972420303681502792767476, 8.919866431381794476886903221783, 9.959803487519015853948514122640
