| L(s) = 1 | − 43.4·2-s + 729·3-s − 6.30e3·4-s − 3.16e4·6-s + 2.00e5·7-s + 6.29e5·8-s + 5.31e5·9-s − 2.68e5·11-s − 4.59e6·12-s − 2.64e7·13-s − 8.71e6·14-s + 2.43e7·16-s − 7.63e7·17-s − 2.30e7·18-s + 1.28e8·19-s + 1.46e8·21-s + 1.16e7·22-s + 7.93e8·23-s + 4.58e8·24-s + 1.14e9·26-s + 3.87e8·27-s − 1.26e9·28-s − 3.73e9·29-s + 5.93e9·31-s − 6.21e9·32-s − 1.95e8·33-s + 3.31e9·34-s + ⋯ |
| L(s) = 1 | − 0.479·2-s + 0.577·3-s − 0.769·4-s − 0.276·6-s + 0.644·7-s + 0.849·8-s + 0.333·9-s − 0.0457·11-s − 0.444·12-s − 1.51·13-s − 0.309·14-s + 0.362·16-s − 0.766·17-s − 0.159·18-s + 0.626·19-s + 0.372·21-s + 0.0219·22-s + 1.11·23-s + 0.490·24-s + 0.728·26-s + 0.192·27-s − 0.496·28-s − 1.16·29-s + 1.20·31-s − 1.02·32-s − 0.0264·33-s + 0.367·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 43.4T + 8.19e3T^{2} \) |
| 7 | \( 1 - 2.00e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 2.68e5T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.64e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 7.63e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.28e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 7.93e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.73e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 5.93e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 5.94e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.56e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.72e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 4.05e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.05e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.80e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 4.06e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 3.65e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 3.50e10T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.05e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.62e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.65e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.09e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 7.65e12T + 6.73e25T^{2} \) |
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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
−11.16662808111416778699204594708, −9.889001554336133086193637880985, −9.133346740422613985855578637788, −8.054839282991928524427318475971, −7.18478765732387810701372031368, −5.16950014699973386825634108432, −4.30191869410640620745525722252, −2.69128721937374815119997366052, −1.34016960436930980795820278921, 0,
1.34016960436930980795820278921, 2.69128721937374815119997366052, 4.30191869410640620745525722252, 5.16950014699973386825634108432, 7.18478765732387810701372031368, 8.054839282991928524427318475971, 9.133346740422613985855578637788, 9.889001554336133086193637880985, 11.16662808111416778699204594708
