L(s) = 1 | − 0.825·2-s − 1.31·4-s + 1.66·5-s − 7-s + 2.73·8-s − 1.37·10-s − 4.60·11-s − 0.261·13-s + 0.825·14-s + 0.379·16-s − 1.30·17-s − 5.83·19-s − 2.20·20-s + 3.79·22-s + 1.54·23-s − 2.21·25-s + 0.215·26-s + 1.31·28-s − 4.76·29-s − 0.627·31-s − 5.79·32-s + 1.08·34-s − 1.66·35-s + 6.11·37-s + 4.81·38-s + 4.57·40-s − 11.5·41-s + ⋯ |
L(s) = 1 | − 0.583·2-s − 0.659·4-s + 0.746·5-s − 0.377·7-s + 0.968·8-s − 0.435·10-s − 1.38·11-s − 0.0724·13-s + 0.220·14-s + 0.0948·16-s − 0.317·17-s − 1.33·19-s − 0.492·20-s + 0.809·22-s + 0.322·23-s − 0.443·25-s + 0.0422·26-s + 0.249·28-s − 0.884·29-s − 0.112·31-s − 1.02·32-s + 0.185·34-s − 0.282·35-s + 1.00·37-s + 0.780·38-s + 0.722·40-s − 1.80·41-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
\(L(1)\) |
\(\approx\) |
\(0.6422825891\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6422825891\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.825T + 2T^{2} \) |
| 5 | \( 1 - 1.66T + 5T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 13 | \( 1 + 0.261T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 + 0.627T + 31T^{2} \) |
| 37 | \( 1 - 6.11T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 4.72T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 - 3.22T + 61T^{2} \) |
| 67 | \( 1 + 5.93T + 67T^{2} \) |
| 71 | \( 1 + 0.659T + 71T^{2} \) |
| 73 | \( 1 + 0.0147T + 73T^{2} \) |
| 79 | \( 1 - 3.62T + 79T^{2} \) |
| 83 | \( 1 + 6.07T + 83T^{2} \) |
| 89 | \( 1 - 0.737T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
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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
−7.83495646614796363722528855164, −7.41102082492570554443328636743, −6.38226910638937749559842130346, −5.78595863213500579228912295822, −5.03609831425612993046735657413, −4.41918977917741107895047872839, −3.49004182163620911260810490571, −2.44914386616414156312855044239, −1.78015029770119791232093805434, −0.42260390319639745142756481829,
0.42260390319639745142756481829, 1.78015029770119791232093805434, 2.44914386616414156312855044239, 3.49004182163620911260810490571, 4.41918977917741107895047872839, 5.03609831425612993046735657413, 5.78595863213500579228912295822, 6.38226910638937749559842130346, 7.41102082492570554443328636743, 7.83495646614796363722528855164