| L(s) = 1 | − 0.843·3-s − 5-s + 4.75·7-s − 2.28·9-s − 3.78·13-s + 0.843·15-s − 4.98·17-s + 3.12·19-s − 4.01·21-s − 9.39·23-s + 25-s + 4.45·27-s + 2.60·29-s + 3.68·31-s − 4.75·35-s − 3.44·37-s + 3.18·39-s − 9.32·41-s − 11.7·43-s + 2.28·45-s + 1.66·47-s + 15.6·49-s + 4.20·51-s − 1.93·53-s − 2.63·57-s − 0.103·59-s + 10.1·61-s + ⋯ |
| L(s) = 1 | − 0.486·3-s − 0.447·5-s + 1.79·7-s − 0.763·9-s − 1.04·13-s + 0.217·15-s − 1.20·17-s + 0.716·19-s − 0.875·21-s − 1.95·23-s + 0.200·25-s + 0.858·27-s + 0.484·29-s + 0.662·31-s − 0.804·35-s − 0.566·37-s + 0.510·39-s − 1.45·41-s − 1.78·43-s + 0.341·45-s + 0.242·47-s + 2.23·49-s + 0.588·51-s − 0.266·53-s − 0.348·57-s − 0.0135·59-s + 1.29·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.192908202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192908202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.843T + 3T^{2} \) |
| 7 | \( 1 - 4.75T + 7T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 9.39T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 1.66T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 + 0.103T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 4.99T + 67T^{2} \) |
| 71 | \( 1 - 6.22T + 71T^{2} \) |
| 73 | \( 1 - 7.30T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 5.69T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.983902429854460633673229961883, −6.94495532630814298602579844057, −6.40835991195165185994967792858, −5.30387211719588335374083818748, −5.05523399582115601879694111005, −4.41415926757215845443079609461, −3.53157972520906985903761836119, −2.38105779858667443147727319724, −1.81978182023038862506030483755, −0.51949558430695909496294226570,
0.51949558430695909496294226570, 1.81978182023038862506030483755, 2.38105779858667443147727319724, 3.53157972520906985903761836119, 4.41415926757215845443079609461, 5.05523399582115601879694111005, 5.30387211719588335374083818748, 6.40835991195165185994967792858, 6.94495532630814298602579844057, 7.983902429854460633673229961883
