| L(s) = 1 | + 2-s + 0.641·3-s + 4-s − 3.42·5-s + 0.641·6-s + 1.47·7-s + 8-s − 2.58·9-s − 3.42·10-s + 0.306·11-s + 0.641·12-s + 1.47·14-s − 2.19·15-s + 16-s + 2.45·17-s − 2.58·18-s + 19-s − 3.42·20-s + 0.948·21-s + 0.306·22-s − 1.35·23-s + 0.641·24-s + 6.74·25-s − 3.58·27-s + 1.47·28-s − 5.10·29-s − 2.19·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.370·3-s + 0.5·4-s − 1.53·5-s + 0.261·6-s + 0.558·7-s + 0.353·8-s − 0.863·9-s − 1.08·10-s + 0.0925·11-s + 0.185·12-s + 0.395·14-s − 0.567·15-s + 0.250·16-s + 0.594·17-s − 0.610·18-s + 0.229·19-s − 0.766·20-s + 0.206·21-s + 0.0654·22-s − 0.283·23-s + 0.130·24-s + 1.34·25-s − 0.689·27-s + 0.279·28-s − 0.947·29-s − 0.401·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.641T + 3T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 - 1.47T + 7T^{2} \) |
| 11 | \( 1 - 0.306T + 11T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 + 5.10T + 29T^{2} \) |
| 31 | \( 1 + 0.214T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 - 5.21T + 41T^{2} \) |
| 43 | \( 1 - 1.40T + 43T^{2} \) |
| 47 | \( 1 - 8.23T + 47T^{2} \) |
| 53 | \( 1 + 0.0519T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 + 7.86T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 + 5.90T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.67872351655427507895535509296, −7.22181456671088823502647482139, −6.07184201075914200327824306009, −5.52433445653315655887759039715, −4.54500785615240455034181389891, −4.06260026186455991304001736767, −3.26927557298093658610742421936, −2.67339365145238606064192970968, −1.40588890725304002389207978212, 0,
1.40588890725304002389207978212, 2.67339365145238606064192970968, 3.26927557298093658610742421936, 4.06260026186455991304001736767, 4.54500785615240455034181389891, 5.52433445653315655887759039715, 6.07184201075914200327824306009, 7.22181456671088823502647482139, 7.67872351655427507895535509296
