| L(s) = 1 | + 9·3-s + 76·5-s + 49·7-s + 81·9-s − 650·11-s + 762·13-s + 684·15-s − 556·17-s + 2.45e3·19-s + 441·21-s + 2.95e3·23-s + 2.65e3·25-s + 729·27-s − 674·29-s + 3.02e3·31-s − 5.85e3·33-s + 3.72e3·35-s + 7.73e3·37-s + 6.85e3·39-s − 1.70e4·41-s − 2.18e4·43-s + 6.15e3·45-s + 2.39e4·47-s + 2.40e3·49-s − 5.00e3·51-s + 1.55e4·53-s − 4.94e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.35·5-s + 0.377·7-s + 1/3·9-s − 1.61·11-s + 1.25·13-s + 0.784·15-s − 0.466·17-s + 1.55·19-s + 0.218·21-s + 1.16·23-s + 0.848·25-s + 0.192·27-s − 0.148·29-s + 0.565·31-s − 0.935·33-s + 0.513·35-s + 0.928·37-s + 0.721·39-s − 1.58·41-s − 1.80·43-s + 0.453·45-s + 1.58·47-s + 1/7·49-s − 0.269·51-s + 0.762·53-s − 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.698389454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.698389454\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 76 T + p^{5} T^{2} \) |
| 11 | \( 1 + 650 T + p^{5} T^{2} \) |
| 13 | \( 1 - 762 T + p^{5} T^{2} \) |
| 17 | \( 1 + 556 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2452 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2950 T + p^{5} T^{2} \) |
| 29 | \( 1 + 674 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3024 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7730 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17016 T + p^{5} T^{2} \) |
| 43 | \( 1 + 21836 T + p^{5} T^{2} \) |
| 47 | \( 1 - 23940 T + p^{5} T^{2} \) |
| 53 | \( 1 - 15594 T + p^{5} T^{2} \) |
| 59 | \( 1 + 5608 T + p^{5} T^{2} \) |
| 61 | \( 1 - 150 T + p^{5} T^{2} \) |
| 67 | \( 1 - 43784 T + p^{5} T^{2} \) |
| 71 | \( 1 - 39178 T + p^{5} T^{2} \) |
| 73 | \( 1 + 23570 T + p^{5} T^{2} \) |
| 79 | \( 1 - 17892 T + p^{5} T^{2} \) |
| 83 | \( 1 + 38972 T + p^{5} T^{2} \) |
| 89 | \( 1 - 6024 T + p^{5} T^{2} \) |
| 97 | \( 1 - 108430 T + p^{5} T^{2} \) |
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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
−10.52620442158443769805143105160, −9.824928876151912114489207941894, −8.864518445677078231993392817966, −8.034428489722437448341799025660, −6.90236336755472082841120908477, −5.68310150337139394211389412150, −4.96600942265442089543271187419, −3.25786949134423230868114304241, −2.26791437222856299287556596058, −1.11090823728191729799376374869,
1.11090823728191729799376374869, 2.26791437222856299287556596058, 3.25786949134423230868114304241, 4.96600942265442089543271187419, 5.68310150337139394211389412150, 6.90236336755472082841120908477, 8.034428489722437448341799025660, 8.864518445677078231993392817966, 9.824928876151912114489207941894, 10.52620442158443769805143105160
