| L(s) = 1 | − 2·3-s − 96·5-s + 49·7-s − 239·9-s − 720·11-s + 572·13-s + 192·15-s + 1.25e3·17-s − 94·19-s − 98·21-s + 96·23-s + 6.09e3·25-s + 964·27-s − 4.37e3·29-s − 6.24e3·31-s + 1.44e3·33-s − 4.70e3·35-s − 1.07e4·37-s − 1.14e3·39-s + 1.20e4·41-s − 9.16e3·43-s + 2.29e4·45-s − 2.58e4·47-s + 2.40e3·49-s − 2.50e3·51-s + 1.01e3·53-s + 6.91e4·55-s + ⋯ |
| L(s) = 1 | − 0.128·3-s − 1.71·5-s + 0.377·7-s − 0.983·9-s − 1.79·11-s + 0.938·13-s + 0.220·15-s + 1.05·17-s − 0.0597·19-s − 0.0484·21-s + 0.0378·23-s + 1.94·25-s + 0.254·27-s − 0.965·29-s − 1.16·31-s + 0.230·33-s − 0.649·35-s − 1.29·37-s − 0.120·39-s + 1.11·41-s − 0.755·43-s + 1.68·45-s − 1.70·47-s + 1/7·49-s − 0.135·51-s + 0.0495·53-s + 3.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 2 T + p^{5} T^{2} \) |
| 5 | \( 1 + 96 T + p^{5} T^{2} \) |
| 11 | \( 1 + 720 T + p^{5} T^{2} \) |
| 13 | \( 1 - 44 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 1254 T + p^{5} T^{2} \) |
| 19 | \( 1 + 94 T + p^{5} T^{2} \) |
| 23 | \( 1 - 96 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4374 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6244 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10798 T + p^{5} T^{2} \) |
| 41 | \( 1 - 12006 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9160 T + p^{5} T^{2} \) |
| 47 | \( 1 + 25836 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1014 T + p^{5} T^{2} \) |
| 59 | \( 1 - 1242 T + p^{5} T^{2} \) |
| 61 | \( 1 - 7592 T + p^{5} T^{2} \) |
| 67 | \( 1 - 41132 T + p^{5} T^{2} \) |
| 71 | \( 1 + 37632 T + p^{5} T^{2} \) |
| 73 | \( 1 + 13438 T + p^{5} T^{2} \) |
| 79 | \( 1 - 6248 T + p^{5} T^{2} \) |
| 83 | \( 1 + 25254 T + p^{5} T^{2} \) |
| 89 | \( 1 + 45126 T + p^{5} T^{2} \) |
| 97 | \( 1 - 107222 T + p^{5} T^{2} \) |
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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
−15.66460879568356725976406438718, −14.56727787606432360931467156002, −12.89076813217567077613700172951, −11.58112929535376601244632812409, −10.75986348281136906360165818499, −8.434082554525968076475551344576, −7.62447871447066040952538120986, −5.31965875775595161984552193376, −3.40240128430237200615070717741, 0,
3.40240128430237200615070717741, 5.31965875775595161984552193376, 7.62447871447066040952538120986, 8.434082554525968076475551344576, 10.75986348281136906360165818499, 11.58112929535376601244632812409, 12.89076813217567077613700172951, 14.56727787606432360931467156002, 15.66460879568356725976406438718
