| L(s) = 1 | − 2-s + 1.01·3-s + 4-s − 0.455·5-s − 1.01·6-s + 0.356·7-s − 8-s − 1.97·9-s + 0.455·10-s + 1.21·11-s + 1.01·12-s − 0.356·14-s − 0.460·15-s + 16-s + 1.03·17-s + 1.97·18-s + 19-s − 0.455·20-s + 0.359·21-s − 1.21·22-s + 8.71·23-s − 1.01·24-s − 4.79·25-s − 5.03·27-s + 0.356·28-s − 3.10·29-s + 0.460·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.583·3-s + 0.5·4-s − 0.203·5-s − 0.412·6-s + 0.134·7-s − 0.353·8-s − 0.659·9-s + 0.143·10-s + 0.366·11-s + 0.291·12-s − 0.0951·14-s − 0.118·15-s + 0.250·16-s + 0.250·17-s + 0.466·18-s + 0.229·19-s − 0.101·20-s + 0.0785·21-s − 0.259·22-s + 1.81·23-s − 0.206·24-s − 0.958·25-s − 0.968·27-s + 0.0672·28-s − 0.576·29-s + 0.0840·30-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 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 + 0.455T + 5T^{2} \) |
| 7 | \( 1 - 0.356T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 23 | \( 1 - 8.71T + 23T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 + 3.69T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.76T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 2.69T + 67T^{2} \) |
| 71 | \( 1 + 1.14T + 71T^{2} \) |
| 73 | \( 1 + 2.88T + 73T^{2} \) |
| 79 | \( 1 + 4.40T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 5.39T + 97T^{2} \) |
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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
−7.75287465902960568819468334766, −7.18952601490311594451571454140, −6.44829564960082940609725736392, −5.56944697178676150370072076155, −4.89281824220870779557343887893, −3.65238143936747123023852381713, −3.20020600945442045858233656650, −2.22332542016260276304098023553, −1.32241036076569215903650005963, 0,
1.32241036076569215903650005963, 2.22332542016260276304098023553, 3.20020600945442045858233656650, 3.65238143936747123023852381713, 4.89281824220870779557343887893, 5.56944697178676150370072076155, 6.44829564960082940609725736392, 7.18952601490311594451571454140, 7.75287465902960568819468334766
