| L(s) = 1 | + 2-s + 4-s + 5-s − 2.37·7-s + 8-s + 10-s − 1.37·11-s + 4.74·13-s − 2.37·14-s + 16-s + 7.37·17-s + 3.37·19-s + 20-s − 1.37·22-s + 4.37·23-s + 25-s + 4.74·26-s − 2.37·28-s + 4.37·29-s − 6.74·31-s + 32-s + 7.37·34-s − 2.37·35-s − 4·37-s + 3.37·38-s + 40-s + 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.896·7-s + 0.353·8-s + 0.316·10-s − 0.413·11-s + 1.31·13-s − 0.634·14-s + 0.250·16-s + 1.78·17-s + 0.773·19-s + 0.223·20-s − 0.292·22-s + 0.911·23-s + 0.200·25-s + 0.930·26-s − 0.448·28-s + 0.811·29-s − 1.21·31-s + 0.176·32-s + 1.26·34-s − 0.400·35-s − 0.657·37-s + 0.547·38-s + 0.158·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.543764488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543764488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 1.62T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 1.37T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 1.62T + 83T^{2} \) |
| 89 | \( 1 - 1.11T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
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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
−10.25993502607387682591666682057, −9.587789180532392215662842182554, −8.546210784056090088603607229462, −7.50631751385539518485989862848, −6.57917823756672287546515249020, −5.77119144029561517694334490406, −5.07163255207739999097331877792, −3.55105060229437274340943099659, −3.05421563973318891113570986544, −1.35239090928621570494829373116,
1.35239090928621570494829373116, 3.05421563973318891113570986544, 3.55105060229437274340943099659, 5.07163255207739999097331877792, 5.77119144029561517694334490406, 6.57917823756672287546515249020, 7.50631751385539518485989862848, 8.546210784056090088603607229462, 9.587789180532392215662842182554, 10.25993502607387682591666682057
