L(s) = 1 | + 2-s + 2.04·3-s + 4-s + 3.81·5-s + 2.04·6-s + 0.856·7-s + 8-s + 1.18·9-s + 3.81·10-s − 0.287·11-s + 2.04·12-s − 2.99·13-s + 0.856·14-s + 7.80·15-s + 16-s + 0.234·17-s + 1.18·18-s − 6.00·19-s + 3.81·20-s + 1.75·21-s − 0.287·22-s + 0.503·23-s + 2.04·24-s + 9.54·25-s − 2.99·26-s − 3.71·27-s + 0.856·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.18·3-s + 0.5·4-s + 1.70·5-s + 0.835·6-s + 0.323·7-s + 0.353·8-s + 0.394·9-s + 1.20·10-s − 0.0868·11-s + 0.590·12-s − 0.829·13-s + 0.229·14-s + 2.01·15-s + 0.250·16-s + 0.0567·17-s + 0.278·18-s − 1.37·19-s + 0.852·20-s + 0.382·21-s − 0.0613·22-s + 0.105·23-s + 0.417·24-s + 1.90·25-s − 0.586·26-s − 0.715·27-s + 0.161·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.930341277\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.930341277\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 7 | \( 1 - 0.856T + 7T^{2} \) |
| 11 | \( 1 + 0.287T + 11T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 17 | \( 1 - 0.234T + 17T^{2} \) |
| 19 | \( 1 + 6.00T + 19T^{2} \) |
| 23 | \( 1 - 0.503T + 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 - 8.72T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 - 0.190T + 43T^{2} \) |
| 47 | \( 1 + 7.85T + 47T^{2} \) |
| 53 | \( 1 - 8.87T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 - 7.01T + 67T^{2} \) |
| 71 | \( 1 - 4.28T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 2.45T + 79T^{2} \) |
| 83 | \( 1 - 9.82T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−9.479281035650972174386306019334, −8.737388491165584220216764691105, −7.962560693145141870573553827022, −6.92813942154898554946612400158, −6.14089575991210090198098325859, −5.30942515141100329346991308412, −4.47002258462638556987917228057, −3.22044099281045178775448601626, −2.33598000808699426505603607165, −1.82029044978213020993903340065,
1.82029044978213020993903340065, 2.33598000808699426505603607165, 3.22044099281045178775448601626, 4.47002258462638556987917228057, 5.30942515141100329346991308412, 6.14089575991210090198098325859, 6.92813942154898554946612400158, 7.962560693145141870573553827022, 8.737388491165584220216764691105, 9.479281035650972174386306019334