| L(s) = 1 | + 2.33·2-s + 3.47·4-s + 0.256·5-s − 3.14·7-s + 3.44·8-s + 0.600·10-s + 11-s − 1.69·13-s − 7.36·14-s + 1.10·16-s − 1.79·17-s + 1.00·19-s + 0.891·20-s + 2.33·22-s − 5.49·23-s − 4.93·25-s − 3.95·26-s − 10.9·28-s + 6.78·29-s − 0.413·31-s − 4.29·32-s − 4.19·34-s − 0.808·35-s − 0.309·37-s + 2.35·38-s + 0.883·40-s − 6.43·41-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.73·4-s + 0.114·5-s − 1.19·7-s + 1.21·8-s + 0.189·10-s + 0.301·11-s − 0.468·13-s − 1.96·14-s + 0.276·16-s − 0.435·17-s + 0.231·19-s + 0.199·20-s + 0.498·22-s − 1.14·23-s − 0.986·25-s − 0.775·26-s − 2.06·28-s + 1.26·29-s − 0.0742·31-s − 0.758·32-s − 0.719·34-s − 0.136·35-s − 0.0508·37-s + 0.382·38-s + 0.139·40-s − 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 5 | \( 1 - 0.256T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 + 5.49T + 23T^{2} \) |
| 29 | \( 1 - 6.78T + 29T^{2} \) |
| 31 | \( 1 + 0.413T + 31T^{2} \) |
| 37 | \( 1 + 0.309T + 37T^{2} \) |
| 41 | \( 1 + 6.43T + 41T^{2} \) |
| 43 | \( 1 - 0.181T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 8.08T + 59T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 9.12T + 71T^{2} \) |
| 73 | \( 1 - 1.63T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 - 19.5T + 97T^{2} \) |
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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
−7.38470709901082437635244721334, −6.60771199044974449617697648088, −6.24747636157577736722077191813, −5.58130341756553169932860797094, −4.72117583057880087140143504788, −4.08226226302083486638318440120, −3.34359564935399581352440187206, −2.71280323044215533832953634354, −1.79254488262065839654605738093, 0,
1.79254488262065839654605738093, 2.71280323044215533832953634354, 3.34359564935399581352440187206, 4.08226226302083486638318440120, 4.72117583057880087140143504788, 5.58130341756553169932860797094, 6.24747636157577736722077191813, 6.60771199044974449617697648088, 7.38470709901082437635244721334
