| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 3·9-s − 6·11-s − 2·13-s + 2·14-s + 16-s + 17-s − 3·18-s − 4·19-s − 6·22-s − 8·23-s − 5·25-s − 2·26-s + 2·28-s + 4·29-s − 4·31-s + 32-s + 34-s − 3·36-s − 2·37-s − 4·38-s − 2·41-s + 8·43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 9-s − 1.80·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.707·18-s − 0.917·19-s − 1.27·22-s − 1.66·23-s − 25-s − 0.392·26-s + 0.377·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 1/2·36-s − 0.328·37-s − 0.648·38-s − 0.312·41-s + 1.21·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.418537259666727600111079570527, −8.007771869805339946200321664903, −7.31107405112762763862775650482, −6.04036888992905934184473721667, −5.53930298457802415693257982558, −4.79904585048628514262019209537, −3.87211422066710201311431646800, −2.65107966939990787725169913995, −2.09060810290794596233433720211, 0,
2.09060810290794596233433720211, 2.65107966939990787725169913995, 3.87211422066710201311431646800, 4.79904585048628514262019209537, 5.53930298457802415693257982558, 6.04036888992905934184473721667, 7.31107405112762763862775650482, 8.007771869805339946200321664903, 8.418537259666727600111079570527
