| L(s) = 1 | − 2-s + 1.80·3-s + 4-s − 5-s − 1.80·6-s + 4.53·7-s − 8-s + 0.273·9-s + 10-s − 11-s + 1.80·12-s + 1.28·13-s − 4.53·14-s − 1.80·15-s + 16-s − 2.49·17-s − 0.273·18-s + 4.30·19-s − 20-s + 8.21·21-s + 22-s − 0.713·23-s − 1.80·24-s + 25-s − 1.28·26-s − 4.93·27-s + 4.53·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.04·3-s + 0.5·4-s − 0.447·5-s − 0.738·6-s + 1.71·7-s − 0.353·8-s + 0.0912·9-s + 0.316·10-s − 0.301·11-s + 0.522·12-s + 0.356·13-s − 1.21·14-s − 0.467·15-s + 0.250·16-s − 0.606·17-s − 0.0644·18-s + 0.987·19-s − 0.223·20-s + 1.79·21-s + 0.213·22-s − 0.148·23-s − 0.369·24-s + 0.200·25-s − 0.252·26-s − 0.949·27-s + 0.857·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.310279263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.310279263\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 1.80T + 3T^{2} \) |
| 7 | \( 1 - 4.53T + 7T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 + 0.713T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 - 6.87T + 31T^{2} \) |
| 37 | \( 1 - 2.58T + 37T^{2} \) |
| 41 | \( 1 - 2.61T + 41T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 - 7.15T + 53T^{2} \) |
| 59 | \( 1 + 2.05T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 - 4.88T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 4.33T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 3.85T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.292953436851440439080476222201, −7.62835152393624973167921883625, −7.50385190795608767834404628488, −6.21401324603192724851426544432, −5.31100348402051851886551767365, −4.48636818453412205789187051538, −3.63707110136755055585157456020, −2.64605146804612419610100810680, −1.96356750447216182688987730020, −0.919206797848796634323582972701,
0.919206797848796634323582972701, 1.96356750447216182688987730020, 2.64605146804612419610100810680, 3.63707110136755055585157456020, 4.48636818453412205789187051538, 5.31100348402051851886551767365, 6.21401324603192724851426544432, 7.50385190795608767834404628488, 7.62835152393624973167921883625, 8.292953436851440439080476222201
