| L(s) = 1 | − 0.732·2-s + 5.92·3-s − 7.46·4-s − 12.8·5-s − 4.33·6-s + 16.9·7-s + 11.3·8-s + 8.14·9-s + 9.41·10-s − 11·11-s − 44.2·12-s + 74.6·13-s − 12.3·14-s − 76.2·15-s + 51.4·16-s − 82.7·17-s − 5.96·18-s − 67.9·19-s + 95.9·20-s + 100.·21-s + 8.05·22-s + 13.3·23-s + 67.1·24-s + 40.2·25-s − 54.6·26-s − 111.·27-s − 126.·28-s + ⋯ |
| L(s) = 1 | − 0.258·2-s + 1.14·3-s − 0.933·4-s − 1.14·5-s − 0.295·6-s + 0.914·7-s + 0.500·8-s + 0.301·9-s + 0.297·10-s − 0.301·11-s − 1.06·12-s + 1.59·13-s − 0.236·14-s − 1.31·15-s + 0.803·16-s − 1.18·17-s − 0.0780·18-s − 0.820·19-s + 1.07·20-s + 1.04·21-s + 0.0780·22-s + 0.121·23-s + 0.570·24-s + 0.322·25-s − 0.412·26-s − 0.796·27-s − 0.852·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8560195510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8560195510\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.732T + 8T^{2} \) |
| 3 | \( 1 - 5.92T + 27T^{2} \) |
| 5 | \( 1 + 12.8T + 125T^{2} \) |
| 7 | \( 1 - 16.9T + 343T^{2} \) |
| 13 | \( 1 - 74.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 65.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 40.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 2.28T + 7.95e4T^{2} \) |
| 47 | \( 1 - 71.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 545.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 101.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 470.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 610.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 978.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 26.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 352.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 847.T + 9.12e5T^{2} \) |
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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
−19.98272888814166802692057074899, −18.95082872831269086040641176420, −17.74965087372963331749051957749, −15.70426635999610925768034563766, −14.45413212802334498598783244085, −13.23394761241722868999602498309, −11.08380472054021309562065233191, −8.738362642763152216189473172240, −8.090000841979261224322862516068, −4.10903776582739696739959352561,
4.10903776582739696739959352561, 8.090000841979261224322862516068, 8.738362642763152216189473172240, 11.08380472054021309562065233191, 13.23394761241722868999602498309, 14.45413212802334498598783244085, 15.70426635999610925768034563766, 17.74965087372963331749051957749, 18.95082872831269086040641176420, 19.98272888814166802692057074899
