L(s) = 1 | − 2.20·2-s − 3.02·3-s + 2.84·4-s − 2.51·5-s + 6.66·6-s + 0.0943·7-s − 1.86·8-s + 6.15·9-s + 5.53·10-s − 11-s − 8.61·12-s − 0.312·13-s − 0.207·14-s + 7.60·15-s − 1.58·16-s + 17-s − 13.5·18-s − 1.98·19-s − 7.15·20-s − 0.285·21-s + 2.20·22-s − 1.69·23-s + 5.65·24-s + 1.31·25-s + 0.687·26-s − 9.52·27-s + 0.268·28-s + ⋯ |
L(s) = 1 | − 1.55·2-s − 1.74·3-s + 1.42·4-s − 1.12·5-s + 2.71·6-s + 0.0356·7-s − 0.660·8-s + 2.05·9-s + 1.75·10-s − 0.301·11-s − 2.48·12-s − 0.0866·13-s − 0.0555·14-s + 1.96·15-s − 0.395·16-s + 0.242·17-s − 3.19·18-s − 0.454·19-s − 1.60·20-s − 0.0622·21-s + 0.469·22-s − 0.352·23-s + 1.15·24-s + 0.263·25-s + 0.134·26-s − 1.83·27-s + 0.0507·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 3 | \( 1 + 3.02T + 3T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 - 0.0943T + 7T^{2} \) |
| 13 | \( 1 + 0.312T + 13T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 - 8.36T + 29T^{2} \) |
| 31 | \( 1 - 0.536T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 47 | \( 1 - 7.01T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 1.64T + 73T^{2} \) |
| 79 | \( 1 + 9.53T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 2.33T + 89T^{2} \) |
| 97 | \( 1 + 1.95T + 97T^{2} \) |
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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
−7.47890400012789035745448097305, −6.98395552034213739460599214375, −6.31780503730726731584659330108, −5.61008884620932720624487924607, −4.59727516342671202380660802357, −4.24473326828943014035517661350, −2.89648420232787314251490735129, −1.62058164303276487746896324393, −0.72149227376235911507374908284, 0,
0.72149227376235911507374908284, 1.62058164303276487746896324393, 2.89648420232787314251490735129, 4.24473326828943014035517661350, 4.59727516342671202380660802357, 5.61008884620932720624487924607, 6.31780503730726731584659330108, 6.98395552034213739460599214375, 7.47890400012789035745448097305