| L(s) = 1 | − 2.70·2-s − 2.86·3-s + 5.31·4-s + 1.46·5-s + 7.74·6-s − 3.90·7-s − 8.97·8-s + 5.20·9-s − 3.94·10-s + 11-s − 15.2·12-s − 2.79·13-s + 10.5·14-s − 4.18·15-s + 13.6·16-s + 17-s − 14.0·18-s + 0.0854·19-s + 7.76·20-s + 11.1·21-s − 2.70·22-s + 2.41·23-s + 25.7·24-s − 2.86·25-s + 7.57·26-s − 6.31·27-s − 20.7·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 1.65·3-s + 2.65·4-s + 0.652·5-s + 3.16·6-s − 1.47·7-s − 3.17·8-s + 1.73·9-s − 1.24·10-s + 0.301·11-s − 4.39·12-s − 0.776·13-s + 2.82·14-s − 1.07·15-s + 3.41·16-s + 0.242·17-s − 3.31·18-s + 0.0196·19-s + 1.73·20-s + 2.43·21-s − 0.576·22-s + 0.502·23-s + 5.24·24-s − 0.573·25-s + 1.48·26-s − 1.21·27-s − 3.92·28-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
| \(L(1)\) |
\(\approx\) |
\(0.2921064741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2921064741\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 + 2.86T + 3T^{2} \) |
| 5 | \( 1 - 1.46T + 5T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 19 | \( 1 - 0.0854T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 + 3.04T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 8.69T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 + 8.25T + 53T^{2} \) |
| 59 | \( 1 + 1.57T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 6.21T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 4.11T + 83T^{2} \) |
| 89 | \( 1 - 2.09T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−7.64714638624591604013090399973, −7.14907279945655656805926316262, −6.41803007516533586441400692382, −6.14617877039431678141955413631, −5.58087328486803296960691377658, −4.42880071492918503287993933455, −3.09804176610815075345154258168, −2.32008866854749202633494869229, −1.18056577490638067622629253609, −0.44967987731499256799935009799,
0.44967987731499256799935009799, 1.18056577490638067622629253609, 2.32008866854749202633494869229, 3.09804176610815075345154258168, 4.42880071492918503287993933455, 5.58087328486803296960691377658, 6.14617877039431678141955413631, 6.41803007516533586441400692382, 7.14907279945655656805926316262, 7.64714638624591604013090399973
