| L(s) = 1 | + 3.19·3-s − 1.16·5-s + 7-s + 7.19·9-s + 5.39·11-s + 1.73·13-s − 3.72·15-s − 1.79·17-s + 19-s + 3.19·21-s − 2.08·23-s − 3.63·25-s + 13.3·27-s − 3.75·29-s + 3.79·31-s + 17.2·33-s − 1.16·35-s + 1.78·37-s + 5.53·39-s + 8.34·41-s − 2.48·43-s − 8.39·45-s + 5.30·47-s + 49-s − 5.71·51-s + 0.355·53-s − 6.30·55-s + ⋯ |
| L(s) = 1 | + 1.84·3-s − 0.522·5-s + 0.377·7-s + 2.39·9-s + 1.62·11-s + 0.480·13-s − 0.962·15-s − 0.434·17-s + 0.229·19-s + 0.696·21-s − 0.435·23-s − 0.727·25-s + 2.57·27-s − 0.696·29-s + 0.680·31-s + 3.00·33-s − 0.197·35-s + 0.292·37-s + 0.885·39-s + 1.30·41-s − 0.379·43-s − 1.25·45-s + 0.774·47-s + 0.142·49-s − 0.800·51-s + 0.0488·53-s − 0.849·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.947261011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.947261011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.19T + 3T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 11 | \( 1 - 5.39T + 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 23 | \( 1 + 2.08T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 - 3.79T + 31T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 + 2.48T + 43T^{2} \) |
| 47 | \( 1 - 5.30T + 47T^{2} \) |
| 53 | \( 1 - 0.355T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 - 7.98T + 67T^{2} \) |
| 71 | \( 1 + 9.55T + 71T^{2} \) |
| 73 | \( 1 + 6.65T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 1.80T + 83T^{2} \) |
| 89 | \( 1 + 7.13T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.83194338444419474201704120677, −7.35605443070910934219327434808, −6.65372417449940990494826517754, −5.83584740499722414748750476563, −4.53192887989612724201959672065, −3.98606358511433435601850117274, −3.64945430738242362554342540131, −2.67033283258827167698856788834, −1.87263005678221448891748804803, −1.08475015464377075407662070246,
1.08475015464377075407662070246, 1.87263005678221448891748804803, 2.67033283258827167698856788834, 3.64945430738242362554342540131, 3.98606358511433435601850117274, 4.53192887989612724201959672065, 5.83584740499722414748750476563, 6.65372417449940990494826517754, 7.35605443070910934219327434808, 7.83194338444419474201704120677
