| L(s) = 1 | − 0.823·2-s − 1.32·4-s + 2.87·5-s + 7-s + 2.73·8-s − 2.37·10-s + 2.49·11-s − 3.14·13-s − 0.823·14-s + 0.390·16-s + 2.04·17-s + 3.80·19-s − 3.80·20-s − 2.05·22-s + 0.178·23-s + 3.28·25-s + 2.58·26-s − 1.32·28-s − 7.77·29-s + 8.24·31-s − 5.79·32-s − 1.68·34-s + 2.87·35-s + 0.0498·37-s − 3.12·38-s + 7.87·40-s + 2.14·41-s + ⋯ |
| L(s) = 1 | − 0.582·2-s − 0.660·4-s + 1.28·5-s + 0.377·7-s + 0.967·8-s − 0.749·10-s + 0.752·11-s − 0.871·13-s − 0.220·14-s + 0.0976·16-s + 0.495·17-s + 0.871·19-s − 0.850·20-s − 0.438·22-s + 0.0372·23-s + 0.656·25-s + 0.507·26-s − 0.249·28-s − 1.44·29-s + 1.48·31-s − 1.02·32-s − 0.288·34-s + 0.486·35-s + 0.00819·37-s − 0.507·38-s + 1.24·40-s + 0.335·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.880205767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.880205767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.823T + 2T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 - 0.178T + 23T^{2} \) |
| 29 | \( 1 + 7.77T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 0.0498T + 37T^{2} \) |
| 41 | \( 1 - 2.14T + 41T^{2} \) |
| 43 | \( 1 - 4.51T + 43T^{2} \) |
| 47 | \( 1 - 0.391T + 47T^{2} \) |
| 53 | \( 1 - 8.31T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 8.25T + 61T^{2} \) |
| 67 | \( 1 + 5.83T + 67T^{2} \) |
| 71 | \( 1 + 8.03T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 5.72T + 79T^{2} \) |
| 83 | \( 1 - 8.13T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 1.62T + 97T^{2} \) |
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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
−7.75078721973351101499165370205, −7.41612738547200905693656005110, −6.41380192690865010432339262149, −5.69259671517140951930693101610, −5.11113966585809370074584057347, −4.43068335933744138393176894823, −3.51600469640004131327516683857, −2.41846283976401927739834411990, −1.60329008823764246808058810278, −0.803309115819676665775444441180,
0.803309115819676665775444441180, 1.60329008823764246808058810278, 2.41846283976401927739834411990, 3.51600469640004131327516683857, 4.43068335933744138393176894823, 5.11113966585809370074584057347, 5.69259671517140951930693101610, 6.41380192690865010432339262149, 7.41612738547200905693656005110, 7.75078721973351101499165370205
