| L(s) = 1 | + 2.59·2-s + 0.697·3-s + 4.72·4-s − 1.05·5-s + 1.80·6-s + 0.620·7-s + 7.05·8-s − 2.51·9-s − 2.72·10-s + 2.93·11-s + 3.29·12-s + 2.63·13-s + 1.60·14-s − 0.733·15-s + 8.83·16-s + 17-s − 6.51·18-s − 4.53·19-s − 4.96·20-s + 0.432·21-s + 7.61·22-s + 3.50·23-s + 4.91·24-s − 3.89·25-s + 6.83·26-s − 3.84·27-s + 2.92·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.402·3-s + 2.36·4-s − 0.470·5-s + 0.737·6-s + 0.234·7-s + 2.49·8-s − 0.837·9-s − 0.862·10-s + 0.885·11-s + 0.949·12-s + 0.731·13-s + 0.429·14-s − 0.189·15-s + 2.20·16-s + 0.242·17-s − 1.53·18-s − 1.04·19-s − 1.11·20-s + 0.0943·21-s + 1.62·22-s + 0.731·23-s + 1.00·24-s − 0.778·25-s + 1.34·26-s − 0.739·27-s + 0.553·28-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
| \(L(1)\) |
\(\approx\) |
\(4.823551442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.823551442\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.59T + 2T^{2} \) |
| 3 | \( 1 - 0.697T + 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 - 0.620T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 - 3.50T + 23T^{2} \) |
| 29 | \( 1 + 7.40T + 29T^{2} \) |
| 31 | \( 1 - 4.66T + 31T^{2} \) |
| 37 | \( 1 + 3.10T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 - 2.18T + 43T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 3.15T + 59T^{2} \) |
| 61 | \( 1 - 3.86T + 61T^{2} \) |
| 67 | \( 1 + 6.80T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 7.11T + 73T^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 - 9.64T + 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−10.89664240141973859764658178247, −9.364048678794522395290214159154, −8.360137398903587913670596013974, −7.48511452055283514248215040187, −6.39611223132731926572582320660, −5.82634784850745442100359014166, −4.68261853031284240793533105614, −3.83614299854164226572239475752, −3.13898562716713524804850785649, −1.87663271766266349497809059787,
1.87663271766266349497809059787, 3.13898562716713524804850785649, 3.83614299854164226572239475752, 4.68261853031284240793533105614, 5.82634784850745442100359014166, 6.39611223132731926572582320660, 7.48511452055283514248215040187, 8.360137398903587913670596013974, 9.364048678794522395290214159154, 10.89664240141973859764658178247
