| L(s) = 1 | + 2-s − 3.07·3-s + 4-s − 3.30·5-s − 3.07·6-s − 2.62·7-s + 8-s + 6.48·9-s − 3.30·10-s + 3.59·11-s − 3.07·12-s − 3.17·13-s − 2.62·14-s + 10.1·15-s + 16-s + 4.85·17-s + 6.48·18-s − 19-s − 3.30·20-s + 8.07·21-s + 3.59·22-s − 3.16·23-s − 3.07·24-s + 5.93·25-s − 3.17·26-s − 10.7·27-s − 2.62·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.77·3-s + 0.5·4-s − 1.47·5-s − 1.25·6-s − 0.990·7-s + 0.353·8-s + 2.16·9-s − 1.04·10-s + 1.08·11-s − 0.888·12-s − 0.879·13-s − 0.700·14-s + 2.62·15-s + 0.250·16-s + 1.17·17-s + 1.52·18-s − 0.229·19-s − 0.739·20-s + 1.76·21-s + 0.766·22-s − 0.660·23-s − 0.628·24-s + 1.18·25-s − 0.622·26-s − 2.06·27-s − 0.495·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5726602054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5726602054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 + 6.15T + 31T^{2} \) |
| 37 | \( 1 - 7.44T + 37T^{2} \) |
| 41 | \( 1 - 0.725T + 41T^{2} \) |
| 43 | \( 1 + 1.68T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 + 9.95T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 5.58T + 67T^{2} \) |
| 71 | \( 1 - 3.62T + 71T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−7.42516246598545169651670517767, −7.01476905813939927283630026235, −6.32725965956562526383376284303, −5.79857630245533194357235148870, −5.03204515348639892729591899058, −4.27348777040628447640018281717, −3.83958897802089496999911159533, −3.05221820785067812586565283312, −1.50018851239033205838018208598, −0.38205663054728895015924175010,
0.38205663054728895015924175010, 1.50018851239033205838018208598, 3.05221820785067812586565283312, 3.83958897802089496999911159533, 4.27348777040628447640018281717, 5.03204515348639892729591899058, 5.79857630245533194357235148870, 6.32725965956562526383376284303, 7.01476905813939927283630026235, 7.42516246598545169651670517767
