| L(s) = 1 | − 152.·2-s + 729·3-s + 1.49e4·4-s − 4.25e4·5-s − 1.10e5·6-s − 5.71e5·7-s − 1.03e6·8-s + 5.31e5·9-s + 6.48e6·10-s − 6.58e6·11-s + 1.09e7·12-s + 4.28e6·13-s + 8.69e7·14-s − 3.10e7·15-s + 3.44e7·16-s − 1.87e8·17-s − 8.08e7·18-s − 2.31e8·19-s − 6.37e8·20-s − 4.16e8·21-s + 1.00e9·22-s + 1.28e9·23-s − 7.52e8·24-s + 5.92e8·25-s − 6.52e8·26-s + 3.87e8·27-s − 8.55e9·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.82·4-s − 1.21·5-s − 0.970·6-s − 1.83·7-s − 1.39·8-s + 0.333·9-s + 2.04·10-s − 1.12·11-s + 1.05·12-s + 0.246·13-s + 3.08·14-s − 0.703·15-s + 0.513·16-s − 1.88·17-s − 0.560·18-s − 1.13·19-s − 2.22·20-s − 1.06·21-s + 1.88·22-s + 1.81·23-s − 0.804·24-s + 0.485·25-s − 0.413·26-s + 0.192·27-s − 3.35·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 729T \) |
| 59 | \( 1 - 4.21e10T \) |
| good | 2 | \( 1 + 152.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 4.25e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.71e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 6.58e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 4.28e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.87e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.31e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.28e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.90e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 1.60e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.91e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.68e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.91e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 2.29e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.99e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 4.98e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 3.43e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.13e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.48e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.64e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.22e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 6.12e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 7.26e11T + 6.73e25T^{2} \) |
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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
−9.614063287223889507770844659732, −8.809946159372137245978627345453, −8.145765358454272451894373754155, −7.05976233800986606701784440658, −6.52007972042882381908814240557, −4.36121222594923517553323856013, −3.07647151600623654293445040679, −2.36996870818998476553097702891, −0.64447348473997649911969013082, 0,
0.64447348473997649911969013082, 2.36996870818998476553097702891, 3.07647151600623654293445040679, 4.36121222594923517553323856013, 6.52007972042882381908814240557, 7.05976233800986606701784440658, 8.145765358454272451894373754155, 8.809946159372137245978627345453, 9.614063287223889507770844659732
