| L(s) = 1 | − 0.739·3-s − 1.43·5-s − 7-s − 2.45·9-s − 4.73·11-s + 1.06·15-s + 2.85·17-s + 0.0123·19-s + 0.739·21-s + 8.75·23-s − 2.93·25-s + 4.03·27-s − 6.55·29-s + 6.65·31-s + 3.49·33-s + 1.43·35-s + 2.21·37-s − 1.13·41-s + 8.89·43-s + 3.52·45-s − 13.0·47-s + 49-s − 2.10·51-s + 6.20·53-s + 6.80·55-s − 0.00909·57-s + 8.21·59-s + ⋯ |
| L(s) = 1 | − 0.426·3-s − 0.642·5-s − 0.377·7-s − 0.817·9-s − 1.42·11-s + 0.274·15-s + 0.692·17-s + 0.00282·19-s + 0.161·21-s + 1.82·23-s − 0.586·25-s + 0.775·27-s − 1.21·29-s + 1.19·31-s + 0.608·33-s + 0.243·35-s + 0.363·37-s − 0.177·41-s + 1.35·43-s + 0.525·45-s − 1.89·47-s + 0.142·49-s − 0.295·51-s + 0.852·53-s + 0.917·55-s − 0.00120·57-s + 1.06·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.739T + 3T^{2} \) |
| 5 | \( 1 + 1.43T + 5T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 19 | \( 1 - 0.0123T + 19T^{2} \) |
| 23 | \( 1 - 8.75T + 23T^{2} \) |
| 29 | \( 1 + 6.55T + 29T^{2} \) |
| 31 | \( 1 - 6.65T + 31T^{2} \) |
| 37 | \( 1 - 2.21T + 37T^{2} \) |
| 41 | \( 1 + 1.13T + 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 8.05T + 61T^{2} \) |
| 67 | \( 1 + 9.19T + 67T^{2} \) |
| 71 | \( 1 - 0.560T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 9.20T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 0.0602T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−7.40796022989503555893215796111, −6.71795505025650699184653822656, −5.84020442008915814252367611118, −5.34628764229688394787549839913, −4.75122183541852275192253017026, −3.72613679224185432812908171200, −3.02185688655325671592827819397, −2.40848820377514508953819720062, −0.914868349433712374937377483106, 0,
0.914868349433712374937377483106, 2.40848820377514508953819720062, 3.02185688655325671592827819397, 3.72613679224185432812908171200, 4.75122183541852275192253017026, 5.34628764229688394787549839913, 5.84020442008915814252367611118, 6.71795505025650699184653822656, 7.40796022989503555893215796111
