| L(s) = 1 | − 9·3-s − 77.6·5-s − 49·7-s + 81·9-s + 477.·11-s − 63.7·13-s + 698.·15-s + 1.03e3·17-s − 667.·19-s + 441·21-s + 3.25e3·23-s + 2.90e3·25-s − 729·27-s + 2.30e3·29-s + 3.71e3·31-s − 4.29e3·33-s + 3.80e3·35-s + 1.22e4·37-s + 573.·39-s − 1.82e3·41-s − 2.07e4·43-s − 6.28e3·45-s − 4.28e3·47-s + 2.40e3·49-s − 9.33e3·51-s + 2.57e4·53-s − 3.70e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.38·5-s − 0.377·7-s + 0.333·9-s + 1.18·11-s − 0.104·13-s + 0.801·15-s + 0.870·17-s − 0.423·19-s + 0.218·21-s + 1.28·23-s + 0.928·25-s − 0.192·27-s + 0.508·29-s + 0.694·31-s − 0.686·33-s + 0.524·35-s + 1.47·37-s + 0.0604·39-s − 0.169·41-s − 1.71·43-s − 0.462·45-s − 0.282·47-s + 0.142·49-s − 0.502·51-s + 1.25·53-s − 1.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.065233244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065233244\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 7 | \( 1 + 49T \) |
| good | 5 | \( 1 + 77.6T + 3.12e3T^{2} \) |
| 11 | \( 1 - 477.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 63.7T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 667.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.83e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.25e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.98e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.35e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.42e5T + 8.58e9T^{2} \) |
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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
−13.04589591097800448108816067438, −11.95372122001901036564881757161, −11.45809232237543543265642907831, −10.11197894600984929543742669939, −8.728896099444783214461133209925, −7.44464416532910332841273569285, −6.37066729040917812049820557988, −4.64553794718278098278065448766, −3.44196985942801103584946040914, −0.812487711873018734811142710361,
0.812487711873018734811142710361, 3.44196985942801103584946040914, 4.64553794718278098278065448766, 6.37066729040917812049820557988, 7.44464416532910332841273569285, 8.728896099444783214461133209925, 10.11197894600984929543742669939, 11.45809232237543543265642907831, 11.95372122001901036564881757161, 13.04589591097800448108816067438
