| L(s) = 1 | + 2-s − 2.28·3-s + 4-s + 4.33·5-s − 2.28·6-s − 0.955·7-s + 8-s + 2.21·9-s + 4.33·10-s − 5.13·11-s − 2.28·12-s − 0.223·13-s − 0.955·14-s − 9.89·15-s + 16-s + 2.89·17-s + 2.21·18-s − 19-s + 4.33·20-s + 2.18·21-s − 5.13·22-s − 3.30·23-s − 2.28·24-s + 13.7·25-s − 0.223·26-s + 1.79·27-s − 0.955·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.31·3-s + 0.5·4-s + 1.93·5-s − 0.932·6-s − 0.361·7-s + 0.353·8-s + 0.737·9-s + 1.37·10-s − 1.54·11-s − 0.659·12-s − 0.0620·13-s − 0.255·14-s − 2.55·15-s + 0.250·16-s + 0.701·17-s + 0.521·18-s − 0.229·19-s + 0.969·20-s + 0.476·21-s − 1.09·22-s − 0.689·23-s − 0.466·24-s + 2.75·25-s − 0.0438·26-s + 0.345·27-s − 0.180·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.485246021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.485246021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.28T + 3T^{2} \) |
| 5 | \( 1 - 4.33T + 5T^{2} \) |
| 7 | \( 1 + 0.955T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 + 0.223T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 + 0.0537T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 + 2.29T + 37T^{2} \) |
| 41 | \( 1 - 0.965T + 41T^{2} \) |
| 43 | \( 1 + 5.82T + 43T^{2} \) |
| 47 | \( 1 - 9.69T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 4.27T + 59T^{2} \) |
| 61 | \( 1 - 1.64T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 - 3.78T + 73T^{2} \) |
| 79 | \( 1 - 9.19T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.52774232229779933191738949797, −6.76848140981102882361467737125, −6.09062941735863093495104293202, −5.77756110186190877756967423654, −5.16611045465621760986757576199, −4.82900030021281993968793221236, −3.47718508368481767651256621509, −2.53760150489885000788331735568, −1.94732994441215179260122903196, −0.73290722076026772522775902025,
0.73290722076026772522775902025, 1.94732994441215179260122903196, 2.53760150489885000788331735568, 3.47718508368481767651256621509, 4.82900030021281993968793221236, 5.16611045465621760986757576199, 5.77756110186190877756967423654, 6.09062941735863093495104293202, 6.76848140981102882361467737125, 7.52774232229779933191738949797
