| L(s) = 1 | − 2.71·2-s − 3-s + 5.34·4-s + 0.130·5-s + 2.71·6-s + 7-s − 9.07·8-s + 9-s − 0.353·10-s + 3.25·11-s − 5.34·12-s + 1.11·13-s − 2.71·14-s − 0.130·15-s + 13.9·16-s + 0.0623·17-s − 2.71·18-s + 2.72·19-s + 0.696·20-s − 21-s − 8.81·22-s + 6.67·23-s + 9.07·24-s − 4.98·25-s − 3.03·26-s − 27-s + 5.34·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.577·3-s + 2.67·4-s + 0.0582·5-s + 1.10·6-s + 0.377·7-s − 3.20·8-s + 0.333·9-s − 0.111·10-s + 0.980·11-s − 1.54·12-s + 0.310·13-s − 0.724·14-s − 0.0336·15-s + 3.47·16-s + 0.0151·17-s − 0.638·18-s + 0.625·19-s + 0.155·20-s − 0.218·21-s − 1.88·22-s + 1.39·23-s + 1.85·24-s − 0.996·25-s − 0.595·26-s − 0.192·27-s + 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8649635803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8649635803\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 5 | \( 1 - 0.130T + 5T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 - 0.0623T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 - 6.67T + 23T^{2} \) |
| 29 | \( 1 + 7.51T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6.34T + 41T^{2} \) |
| 43 | \( 1 - 2.41T + 43T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 + 6.90T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 + 3.59T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 3.03T + 71T^{2} \) |
| 73 | \( 1 + 0.955T + 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 - 0.768T + 83T^{2} \) |
| 89 | \( 1 - 2.54T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−7.82514139705197947087019632125, −7.35621662052587424339943402787, −6.63963396001587820186971361288, −6.09580826841409756299468646942, −5.36980686640813817431273509815, −4.22174650364660967271046116429, −3.21667770708085207181179749273, −2.22282082622818646823834177452, −1.32455843120321185736160464570, −0.72415550687632770226451413651,
0.72415550687632770226451413651, 1.32455843120321185736160464570, 2.22282082622818646823834177452, 3.21667770708085207181179749273, 4.22174650364660967271046116429, 5.36980686640813817431273509815, 6.09580826841409756299468646942, 6.63963396001587820186971361288, 7.35621662052587424339943402787, 7.82514139705197947087019632125
