| L(s) = 1 | + 2-s + 3.14·3-s + 4-s + 3.14·6-s − 3.50·7-s + 8-s + 6.86·9-s + 2·11-s + 3.14·12-s + 3.14·13-s − 3.50·14-s + 16-s + 17-s + 6.86·18-s − 3.86·19-s − 11.0·21-s + 2·22-s − 7.50·23-s + 3.14·24-s + 3.14·26-s + 12.1·27-s − 3.50·28-s − 0.646·29-s − 3.91·31-s + 32-s + 6.28·33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.81·3-s + 0.5·4-s + 1.28·6-s − 1.32·7-s + 0.353·8-s + 2.28·9-s + 0.603·11-s + 0.906·12-s + 0.871·13-s − 0.936·14-s + 0.250·16-s + 0.242·17-s + 1.61·18-s − 0.887·19-s − 2.40·21-s + 0.426·22-s − 1.56·23-s + 0.641·24-s + 0.616·26-s + 2.33·27-s − 0.662·28-s − 0.119·29-s − 0.703·31-s + 0.176·32-s + 1.09·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.984816435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.984816435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 + 7.50T + 23T^{2} \) |
| 29 | \( 1 + 0.646T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 - 8.95T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 + 0.910T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 3.19T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 + 4.41T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−9.847063256840800855330274455298, −9.425653968846041235315771901174, −8.452569477548514819489799541518, −7.74619841553949193615512781314, −6.65751955612563700695125631879, −6.05916334614714507653136418935, −4.27023455897007528115742610244, −3.68029669773616614499017300039, −2.91605479924809469648722584498, −1.79903805494525743817576987479,
1.79903805494525743817576987479, 2.91605479924809469648722584498, 3.68029669773616614499017300039, 4.27023455897007528115742610244, 6.05916334614714507653136418935, 6.65751955612563700695125631879, 7.74619841553949193615512781314, 8.452569477548514819489799541518, 9.425653968846041235315771901174, 9.847063256840800855330274455298
