| L(s) = 1 | + 1.17·2-s − 3.05·3-s − 0.628·4-s − 2.22·5-s − 3.57·6-s − 4.41·7-s − 3.07·8-s + 6.32·9-s − 2.61·10-s − 0.348·11-s + 1.91·12-s + 0.726·13-s − 5.16·14-s + 6.80·15-s − 2.34·16-s − 17-s + 7.40·18-s + 2.15·19-s + 1.40·20-s + 13.4·21-s − 0.408·22-s − 4.05·23-s + 9.39·24-s − 0.0291·25-s + 0.851·26-s − 10.1·27-s + 2.77·28-s + ⋯ |
| L(s) = 1 | + 0.828·2-s − 1.76·3-s − 0.314·4-s − 0.997·5-s − 1.46·6-s − 1.66·7-s − 1.08·8-s + 2.10·9-s − 0.825·10-s − 0.105·11-s + 0.553·12-s + 0.201·13-s − 1.38·14-s + 1.75·15-s − 0.587·16-s − 0.242·17-s + 1.74·18-s + 0.494·19-s + 0.313·20-s + 2.94·21-s − 0.0870·22-s − 0.844·23-s + 1.91·24-s − 0.00582·25-s + 0.166·26-s − 1.95·27-s + 0.523·28-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
| \(L(1)\) |
\(\approx\) |
\(0.3534417387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3534417387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.17T + 2T^{2} \) |
| 3 | \( 1 + 3.05T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 0.348T + 11T^{2} \) |
| 13 | \( 1 - 0.726T + 13T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 - 5.22T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 - 2.87T + 41T^{2} \) |
| 43 | \( 1 + 7.45T + 43T^{2} \) |
| 53 | \( 1 + 1.33T + 53T^{2} \) |
| 59 | \( 1 - 5.50T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 + 5.57T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 9.03T + 79T^{2} \) |
| 83 | \( 1 + 1.28T + 83T^{2} \) |
| 89 | \( 1 + 5.39T + 89T^{2} \) |
| 97 | \( 1 - 6.95T + 97T^{2} \) |
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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
−10.29611067848116807514791181830, −9.800933151127434773651596265088, −8.594056330611523557129297320478, −7.25118954365955622432665340697, −6.39748559014953438258120594645, −5.90105050595134222051697281756, −4.87864754258680624486683874311, −4.06789671994534579095187154615, −3.21769205165245721482548422859, −0.44827964622044045438674547544,
0.44827964622044045438674547544, 3.21769205165245721482548422859, 4.06789671994534579095187154615, 4.87864754258680624486683874311, 5.90105050595134222051697281756, 6.39748559014953438258120594645, 7.25118954365955622432665340697, 8.594056330611523557129297320478, 9.800933151127434773651596265088, 10.29611067848116807514791181830
