| L(s) = 1 | − 2-s − 1.14·3-s + 4-s − 0.568·5-s + 1.14·6-s − 8-s − 1.68·9-s + 0.568·10-s + 0.961·11-s − 1.14·12-s + 2.99·13-s + 0.650·15-s + 16-s − 4.71·17-s + 1.68·18-s − 3.44·19-s − 0.568·20-s − 0.961·22-s + 6.53·23-s + 1.14·24-s − 4.67·25-s − 2.99·26-s + 5.37·27-s + 0.291·29-s − 0.650·30-s − 2.50·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.661·3-s + 0.5·4-s − 0.254·5-s + 0.467·6-s − 0.353·8-s − 0.562·9-s + 0.179·10-s + 0.289·11-s − 0.330·12-s + 0.831·13-s + 0.168·15-s + 0.250·16-s − 1.14·17-s + 0.397·18-s − 0.790·19-s − 0.127·20-s − 0.205·22-s + 1.36·23-s + 0.233·24-s − 0.935·25-s − 0.587·26-s + 1.03·27-s + 0.0542·29-s − 0.118·30-s − 0.449·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 1.14T + 3T^{2} \) |
| 5 | \( 1 + 0.568T + 5T^{2} \) |
| 11 | \( 1 - 0.961T + 11T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 + 3.44T + 19T^{2} \) |
| 23 | \( 1 - 6.53T + 23T^{2} \) |
| 29 | \( 1 - 0.291T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 + 2.16T + 47T^{2} \) |
| 53 | \( 1 - 0.270T + 53T^{2} \) |
| 59 | \( 1 + 1.11T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 + 7.13T + 67T^{2} \) |
| 71 | \( 1 - 4.20T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 83 | \( 1 - 3.34T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.61276763122296787972759384114, −6.64244099970729882244844149028, −6.34622309831612678043427098089, −5.60921181455239849539234669747, −4.72295530637618256477707707515, −3.94549081399570385180773201052, −2.99593632357905173661865869202, −2.10202708274896386654759128057, −0.999879008803337345139322158273, 0,
0.999879008803337345139322158273, 2.10202708274896386654759128057, 2.99593632357905173661865869202, 3.94549081399570385180773201052, 4.72295530637618256477707707515, 5.60921181455239849539234669747, 6.34622309831612678043427098089, 6.64244099970729882244844149028, 7.61276763122296787972759384114
