| L(s) = 1 | + 2-s + 0.462·3-s + 4-s + 1.39·5-s + 0.462·6-s − 3.18·7-s + 8-s − 2.78·9-s + 1.39·10-s + 0.323·11-s + 0.462·12-s + 2.32·13-s − 3.18·14-s + 0.646·15-s + 16-s + 1.72·17-s − 2.78·18-s − 2.86·19-s + 1.39·20-s − 1.47·21-s + 0.323·22-s − 6.50·23-s + 0.462·24-s − 3.04·25-s + 2.32·26-s − 2.67·27-s − 3.18·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.267·3-s + 0.5·4-s + 0.625·5-s + 0.188·6-s − 1.20·7-s + 0.353·8-s − 0.928·9-s + 0.442·10-s + 0.0975·11-s + 0.133·12-s + 0.644·13-s − 0.851·14-s + 0.167·15-s + 0.250·16-s + 0.417·17-s − 0.656·18-s − 0.656·19-s + 0.312·20-s − 0.321·21-s + 0.0689·22-s − 1.35·23-s + 0.0944·24-s − 0.609·25-s + 0.455·26-s − 0.515·27-s − 0.601·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.581070188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581070188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 - 0.462T + 3T^{2} \) |
| 5 | \( 1 - 1.39T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 0.323T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 + 6.50T + 23T^{2} \) |
| 29 | \( 1 - 0.0643T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 4.79T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.60T + 59T^{2} \) |
| 67 | \( 1 + 4.69T + 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 4.19T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 2.51T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.56636525188667121607541374278, −12.63528687892073181496647482021, −11.58227983111631737057892631779, −10.30512615411416245655845002569, −9.340890118945758451597297700292, −8.037056548278346649509330033227, −6.36138462219540663594656656337, −5.80455678809977194165861904382, −3.89655477077336814827851751078, −2.57695791663702807327213774399,
2.57695791663702807327213774399, 3.89655477077336814827851751078, 5.80455678809977194165861904382, 6.36138462219540663594656656337, 8.037056548278346649509330033227, 9.340890118945758451597297700292, 10.30512615411416245655845002569, 11.58227983111631737057892631779, 12.63528687892073181496647482021, 13.56636525188667121607541374278
