| L(s) = 1 | − 2-s − 1.93·3-s + 4-s + 1.93·5-s + 1.93·6-s + 1.50·7-s − 8-s + 0.745·9-s − 1.93·10-s + 0.745·11-s − 1.93·12-s + 0.427·13-s − 1.50·14-s − 3.74·15-s + 16-s − 17-s − 0.745·18-s − 3.50·19-s + 1.93·20-s − 2.91·21-s − 0.745·22-s − 3.36·23-s + 1.93·24-s − 1.25·25-s − 0.427·26-s + 4.36·27-s + 1.50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.865·5-s + 0.790·6-s + 0.570·7-s − 0.353·8-s + 0.248·9-s − 0.612·10-s + 0.224·11-s − 0.558·12-s + 0.118·13-s − 0.403·14-s − 0.967·15-s + 0.250·16-s − 0.242·17-s − 0.175·18-s − 0.804·19-s + 0.432·20-s − 0.636·21-s − 0.159·22-s − 0.701·23-s + 0.395·24-s − 0.250·25-s − 0.0837·26-s + 0.839·27-s + 0.285·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 - 1.93T + 5T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 - 0.745T + 11T^{2} \) |
| 13 | \( 1 - 0.427T + 13T^{2} \) |
| 19 | \( 1 + 3.50T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + 6.17T + 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 37 | \( 1 + 7.44T + 37T^{2} \) |
| 41 | \( 1 + 5.69T + 41T^{2} \) |
| 43 | \( 1 + 8.18T + 43T^{2} \) |
| 47 | \( 1 - 6.69T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 61 | \( 1 - 3.20T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 + 9.55T + 73T^{2} \) |
| 79 | \( 1 - 3.80T + 79T^{2} \) |
| 83 | \( 1 + 0.745T + 83T^{2} \) |
| 89 | \( 1 + 1.81T + 89T^{2} \) |
| 97 | \( 1 - 6.27T + 97T^{2} \) |
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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
−8.752306350811317089585682241928, −8.154404551357667837589339753750, −7.01756280276192793457543068048, −6.38978151522755775923262378650, −5.67226573043648165482404113016, −5.04150096926204498538260831968, −3.84878406652379894506263852138, −2.32409806579437273135988957733, −1.47052493605156283858682293210, 0,
1.47052493605156283858682293210, 2.32409806579437273135988957733, 3.84878406652379894506263852138, 5.04150096926204498538260831968, 5.67226573043648165482404113016, 6.38978151522755775923262378650, 7.01756280276192793457543068048, 8.154404551357667837589339753750, 8.752306350811317089585682241928
