| L(s) = 1 | + 0.530·2-s + 3.23·3-s − 1.71·4-s + 4.03·5-s + 1.71·6-s + 0.377·7-s − 1.97·8-s + 7.47·9-s + 2.14·10-s − 0.339·11-s − 5.56·12-s − 3.39·13-s + 0.200·14-s + 13.0·15-s + 2.39·16-s − 17-s + 3.96·18-s − 7.29·19-s − 6.94·20-s + 1.22·21-s − 0.180·22-s − 1.78·23-s − 6.38·24-s + 11.3·25-s − 1.79·26-s + 14.4·27-s − 0.649·28-s + ⋯ |
| L(s) = 1 | + 0.374·2-s + 1.86·3-s − 0.859·4-s + 1.80·5-s + 0.700·6-s + 0.142·7-s − 0.696·8-s + 2.49·9-s + 0.677·10-s − 0.102·11-s − 1.60·12-s − 0.940·13-s + 0.0535·14-s + 3.37·15-s + 0.598·16-s − 0.242·17-s + 0.934·18-s − 1.67·19-s − 1.55·20-s + 0.266·21-s − 0.0383·22-s − 0.373·23-s − 1.30·24-s + 2.26·25-s − 0.352·26-s + 2.79·27-s − 0.122·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.550426524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.550426524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.530T + 2T^{2} \) |
| 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 7 | \( 1 - 0.377T + 7T^{2} \) |
| 11 | \( 1 + 0.339T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 19 | \( 1 + 7.29T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 6.42T + 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 - 7.91T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 1.53T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 0.449T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−9.857512007152383067113503578175, −9.326631010884181284792706307710, −8.843660393338515887121263472478, −7.962603948857564845227666216614, −6.85348958251307790262668867156, −5.72433314568185472341404949344, −4.71478160719479798369943156257, −3.75995733684252975706860719127, −2.51233920797145294415684445820, −1.89967042105386312037316816052,
1.89967042105386312037316816052, 2.51233920797145294415684445820, 3.75995733684252975706860719127, 4.71478160719479798369943156257, 5.72433314568185472341404949344, 6.85348958251307790262668867156, 7.962603948857564845227666216614, 8.843660393338515887121263472478, 9.326631010884181284792706307710, 9.857512007152383067113503578175
