| L(s) = 1 | − 8.44·2-s + 28.1·3-s + 39.2·4-s + 25·5-s − 237.·6-s − 83.6·7-s − 61.3·8-s + 550.·9-s − 211.·10-s + 87.3·11-s + 1.10e3·12-s + 920.·13-s + 705.·14-s + 704.·15-s − 738.·16-s − 712.·17-s − 4.65e3·18-s + 728.·19-s + 981.·20-s − 2.35e3·21-s − 737.·22-s − 529·23-s − 1.72e3·24-s + 625·25-s − 7.77e3·26-s + 8.67e3·27-s − 3.28e3·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.80·3-s + 1.22·4-s + 0.447·5-s − 2.69·6-s − 0.645·7-s − 0.338·8-s + 2.26·9-s − 0.667·10-s + 0.217·11-s + 2.21·12-s + 1.51·13-s + 0.962·14-s + 0.808·15-s − 0.721·16-s − 0.598·17-s − 3.38·18-s + 0.463·19-s + 0.548·20-s − 1.16·21-s − 0.324·22-s − 0.208·23-s − 0.612·24-s + 0.200·25-s − 2.25·26-s + 2.28·27-s − 0.791·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.872486200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.872486200\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 8.44T + 32T^{2} \) |
| 3 | \( 1 - 28.1T + 243T^{2} \) |
| 7 | \( 1 + 83.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 87.3T + 1.61e5T^{2} \) |
| 13 | \( 1 - 920.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 712.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 728.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 1.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 536.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.54e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.08e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.45e5T + 8.58e9T^{2} \) |
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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
−13.01625713524376543774342482654, −11.06955140584552353143090490522, −9.933635539456683194154542764298, −9.229292674333859529923699440501, −8.608302820811833076677491374338, −7.65009556931469228593669870368, −6.49489885265259908547923857644, −3.86312064518784024257907871651, −2.47420444457956940407434633430, −1.21980223088510120628662331946,
1.21980223088510120628662331946, 2.47420444457956940407434633430, 3.86312064518784024257907871651, 6.49489885265259908547923857644, 7.65009556931469228593669870368, 8.608302820811833076677491374338, 9.229292674333859529923699440501, 9.933635539456683194154542764298, 11.06955140584552353143090490522, 13.01625713524376543774342482654
