| L(s) = 1 | + 1.47·2-s + 3-s + 0.182·4-s + 5-s + 1.47·6-s + 2.29·7-s − 2.68·8-s + 9-s + 1.47·10-s + 0.182·12-s + 2.14·13-s + 3.39·14-s + 15-s − 4.33·16-s − 0.544·17-s + 1.47·18-s + 2.18·19-s + 0.182·20-s + 2.29·21-s + 2.03·23-s − 2.68·24-s + 25-s + 3.16·26-s + 27-s + 0.418·28-s + 9.94·29-s + 1.47·30-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.577·3-s + 0.0911·4-s + 0.447·5-s + 0.603·6-s + 0.867·7-s − 0.949·8-s + 0.333·9-s + 0.467·10-s + 0.0526·12-s + 0.593·13-s + 0.906·14-s + 0.258·15-s − 1.08·16-s − 0.132·17-s + 0.348·18-s + 0.500·19-s + 0.0407·20-s + 0.500·21-s + 0.425·23-s − 0.548·24-s + 0.200·25-s + 0.620·26-s + 0.192·27-s + 0.0790·28-s + 1.84·29-s + 0.269·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.985539922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.985539922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 7 | \( 1 - 2.29T + 7T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 + 0.544T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 - 2.03T + 23T^{2} \) |
| 29 | \( 1 - 9.94T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 - 0.620T + 43T^{2} \) |
| 47 | \( 1 + 0.378T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 8.07T + 59T^{2} \) |
| 61 | \( 1 + 8.72T + 61T^{2} \) |
| 67 | \( 1 + 9.75T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 7.85T + 73T^{2} \) |
| 79 | \( 1 - 9.22T + 79T^{2} \) |
| 83 | \( 1 - 9.45T + 83T^{2} \) |
| 89 | \( 1 - 0.583T + 89T^{2} \) |
| 97 | \( 1 + 5.37T + 97T^{2} \) |
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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
−8.996771396087182805863567950655, −8.646548466705989431249614376016, −7.72430851673907460203923360645, −6.66678283705470563205492839797, −5.92319534438457257042271176238, −4.94081810450433488104508055153, −4.47558276610164603896235502731, −3.37459815923837553501007013338, −2.59950914380238059770492480143, −1.29220227461572524215152317490,
1.29220227461572524215152317490, 2.59950914380238059770492480143, 3.37459815923837553501007013338, 4.47558276610164603896235502731, 4.94081810450433488104508055153, 5.92319534438457257042271176238, 6.66678283705470563205492839797, 7.72430851673907460203923360645, 8.646548466705989431249614376016, 8.996771396087182805863567950655
