| L(s) = 1 | + 0.684·2-s + 1.13·3-s − 1.53·4-s − 5-s + 0.773·6-s − 2.41·8-s − 1.72·9-s − 0.684·10-s + 2.13·11-s − 1.73·12-s + 1.72·13-s − 1.13·15-s + 1.40·16-s − 2.34·17-s − 1.17·18-s − 0.298·19-s + 1.53·20-s + 1.46·22-s + 3.90·23-s − 2.73·24-s + 25-s + 1.18·26-s − 5.33·27-s − 5.60·29-s − 0.773·30-s + 8.19·31-s + 5.79·32-s + ⋯ |
| L(s) = 1 | + 0.483·2-s + 0.652·3-s − 0.765·4-s − 0.447·5-s + 0.315·6-s − 0.854·8-s − 0.574·9-s − 0.216·10-s + 0.644·11-s − 0.499·12-s + 0.478·13-s − 0.291·15-s + 0.352·16-s − 0.567·17-s − 0.277·18-s − 0.0684·19-s + 0.342·20-s + 0.311·22-s + 0.814·23-s − 0.557·24-s + 0.200·25-s + 0.231·26-s − 1.02·27-s − 1.04·29-s − 0.141·30-s + 1.47·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 0.684T + 2T^{2} \) |
| 3 | \( 1 - 1.13T + 3T^{2} \) |
| 11 | \( 1 - 2.13T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 + 0.298T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 0.959T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 - 8.96T + 73T^{2} \) |
| 79 | \( 1 - 0.797T + 79T^{2} \) |
| 83 | \( 1 + 8.88T + 83T^{2} \) |
| 89 | \( 1 + 4.36T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.46926777704296769561780174653, −6.64292059555560593599631081969, −5.93592619634514445134239428086, −5.24114136998234719840707956907, −4.41736789284279404743986211619, −3.86028791957426015660473213124, −3.21829549696445435562678400947, −2.49553856383672395189972067670, −1.20793333373620291661931877571, 0,
1.20793333373620291661931877571, 2.49553856383672395189972067670, 3.21829549696445435562678400947, 3.86028791957426015660473213124, 4.41736789284279404743986211619, 5.24114136998234719840707956907, 5.93592619634514445134239428086, 6.64292059555560593599631081969, 7.46926777704296769561780174653
