| L(s) = 1 | + 0.841·3-s + 5-s + 4.75·7-s − 2.29·9-s + 3.91·11-s + 13-s + 0.841·15-s + 6·17-s + 3.91·19-s + 3.99·21-s + 6.97·23-s + 25-s − 4.45·27-s − 5.29·29-s − 5.59·31-s + 3.29·33-s + 4.75·35-s + 1.29·37-s + 0.841·39-s − 8.58·41-s + 2.52·43-s − 2.29·45-s + 4.75·47-s + 15.5·49-s + 5.05·51-s − 12.5·53-s + 3.91·55-s + ⋯ |
| L(s) = 1 | + 0.485·3-s + 0.447·5-s + 1.79·7-s − 0.763·9-s + 1.17·11-s + 0.277·13-s + 0.217·15-s + 1.45·17-s + 0.897·19-s + 0.872·21-s + 1.45·23-s + 0.200·25-s − 0.857·27-s − 0.982·29-s − 1.00·31-s + 0.572·33-s + 0.803·35-s + 0.212·37-s + 0.134·39-s − 1.34·41-s + 0.385·43-s − 0.341·45-s + 0.693·47-s + 2.22·49-s + 0.707·51-s − 1.72·53-s + 0.527·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.541583441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.541583441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.841T + 3T^{2} \) |
| 7 | \( 1 - 4.75T + 7T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 - 6.97T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 - 1.29T + 37T^{2} \) |
| 41 | \( 1 + 8.58T + 41T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 - 4.75T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 7.27T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 3.06T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 9.29T + 73T^{2} \) |
| 79 | \( 1 + 7.82T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.402841412010533252162852532560, −7.74044483963399220957477946571, −7.21029373866508554021089151953, −6.03487488681387421024818226974, −5.39829013796956619678085570557, −4.80278733411239058468275421810, −3.68773477119110878756689717884, −3.00487222002321927227799271693, −1.73890657273346339745969879906, −1.23059180933521235920959965148,
1.23059180933521235920959965148, 1.73890657273346339745969879906, 3.00487222002321927227799271693, 3.68773477119110878756689717884, 4.80278733411239058468275421810, 5.39829013796956619678085570557, 6.03487488681387421024818226974, 7.21029373866508554021089151953, 7.74044483963399220957477946571, 8.402841412010533252162852532560
