| L(s) = 1 | − 2-s + 1.87·3-s + 4-s + 1.89·5-s − 1.87·6-s − 3.86·7-s − 8-s + 0.512·9-s − 1.89·10-s + 11-s + 1.87·12-s − 3.71·13-s + 3.86·14-s + 3.56·15-s + 16-s − 1.72·17-s − 0.512·18-s + 1.89·20-s − 7.23·21-s − 22-s + 7.66·23-s − 1.87·24-s − 1.39·25-s + 3.71·26-s − 4.66·27-s − 3.86·28-s + 4.54·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.08·3-s + 0.5·4-s + 0.849·5-s − 0.765·6-s − 1.45·7-s − 0.353·8-s + 0.170·9-s − 0.600·10-s + 0.301·11-s + 0.541·12-s − 1.02·13-s + 1.03·14-s + 0.919·15-s + 0.250·16-s − 0.417·17-s − 0.120·18-s + 0.424·20-s − 1.57·21-s − 0.213·22-s + 1.59·23-s − 0.382·24-s − 0.278·25-s + 0.727·26-s − 0.897·27-s − 0.729·28-s + 0.844·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.87T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 23 | \( 1 - 7.66T + 23T^{2} \) |
| 29 | \( 1 - 4.54T + 29T^{2} \) |
| 31 | \( 1 - 1.14T + 31T^{2} \) |
| 37 | \( 1 - 5.00T + 37T^{2} \) |
| 41 | \( 1 + 0.967T + 41T^{2} \) |
| 43 | \( 1 - 0.985T + 43T^{2} \) |
| 47 | \( 1 - 2.49T + 47T^{2} \) |
| 53 | \( 1 + 0.759T + 53T^{2} \) |
| 59 | \( 1 + 4.46T + 59T^{2} \) |
| 61 | \( 1 - 1.79T + 61T^{2} \) |
| 67 | \( 1 + 0.118T + 67T^{2} \) |
| 71 | \( 1 - 4.79T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 + 1.27T + 89T^{2} \) |
| 97 | \( 1 + 4.17T + 97T^{2} \) |
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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
−7.45977950274613877815477090479, −6.96937177472257454283837198426, −6.28138060338476824734360949545, −5.65327222729608127427761906507, −4.59698765787006320292006724309, −3.55674320441298911568025238241, −2.66813141813958745088633687272, −2.59283103409630112351354673377, −1.33964713943038356670309041203, 0,
1.33964713943038356670309041203, 2.59283103409630112351354673377, 2.66813141813958745088633687272, 3.55674320441298911568025238241, 4.59698765787006320292006724309, 5.65327222729608127427761906507, 6.28138060338476824734360949545, 6.96937177472257454283837198426, 7.45977950274613877815477090479
