| L(s) = 1 | − 1.89·2-s + 2.39·3-s + 1.60·4-s + 2.00·5-s − 4.54·6-s + 2.01·7-s + 0.750·8-s + 2.72·9-s − 3.81·10-s + 0.737·11-s + 3.84·12-s + 4.77·13-s − 3.83·14-s + 4.80·15-s − 4.63·16-s − 17-s − 5.17·18-s − 3.77·19-s + 3.22·20-s + 4.83·21-s − 1.40·22-s + 7.57·23-s + 1.79·24-s − 0.961·25-s − 9.07·26-s − 0.651·27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 1.38·3-s + 0.802·4-s + 0.898·5-s − 1.85·6-s + 0.762·7-s + 0.265·8-s + 0.909·9-s − 1.20·10-s + 0.222·11-s + 1.10·12-s + 1.32·13-s − 1.02·14-s + 1.24·15-s − 1.15·16-s − 0.242·17-s − 1.22·18-s − 0.865·19-s + 0.721·20-s + 1.05·21-s − 0.298·22-s + 1.57·23-s + 0.366·24-s − 0.192·25-s − 1.77·26-s − 0.125·27-s + 0.612·28-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
| \(L(1)\) |
\(\approx\) |
\(1.620330670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620330670\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.89T + 2T^{2} \) |
| 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 - 2.00T + 5T^{2} \) |
| 7 | \( 1 - 2.01T + 7T^{2} \) |
| 11 | \( 1 - 0.737T + 11T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 19 | \( 1 + 3.77T + 19T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 + 2.66T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 - 1.27T + 37T^{2} \) |
| 41 | \( 1 + 9.76T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 53 | \( 1 - 8.28T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 2.00T + 61T^{2} \) |
| 67 | \( 1 + 6.97T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 3.28T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 7.22T + 97T^{2} \) |
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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
−9.977777495087390064950526564051, −9.066043507598076356554550794592, −8.780642822102628473101464777929, −8.118875759191206645253123134680, −7.22998178552246332229379050879, −6.19573865257260795029315743935, −4.78901062182190281831204063296, −3.50866321653998641182135900542, −2.14657837262786979236568021004, −1.43823011988455158472193410295,
1.43823011988455158472193410295, 2.14657837262786979236568021004, 3.50866321653998641182135900542, 4.78901062182190281831204063296, 6.19573865257260795029315743935, 7.22998178552246332229379050879, 8.118875759191206645253123134680, 8.780642822102628473101464777929, 9.066043507598076356554550794592, 9.977777495087390064950526564051
