| L(s) = 1 | + 143.·2-s − 729·3-s + 1.23e4·4-s − 5.16e4·5-s − 1.04e5·6-s − 5.73e5·7-s + 5.89e5·8-s + 5.31e5·9-s − 7.39e6·10-s + 1.13e7·11-s − 8.97e6·12-s + 1.95e7·13-s − 8.21e7·14-s + 3.76e7·15-s − 1.64e7·16-s + 4.63e7·17-s + 7.60e7·18-s + 3.60e8·19-s − 6.36e8·20-s + 4.18e8·21-s + 1.62e9·22-s − 1.09e9·23-s − 4.29e8·24-s + 1.44e9·25-s + 2.80e9·26-s − 3.87e8·27-s − 7.06e9·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.50·4-s − 1.47·5-s − 0.913·6-s − 1.84·7-s + 0.794·8-s + 0.333·9-s − 2.33·10-s + 1.92·11-s − 0.867·12-s + 1.12·13-s − 2.91·14-s + 0.853·15-s − 0.244·16-s + 0.465·17-s + 0.527·18-s + 1.75·19-s − 2.22·20-s + 1.06·21-s + 3.04·22-s − 1.53·23-s − 0.458·24-s + 1.18·25-s + 1.77·26-s − 0.192·27-s − 2.77·28-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 | \( 1 - 143.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 5.16e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.73e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.13e7T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.95e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 4.63e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 3.60e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.09e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 5.94e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 4.82e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.53e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.07e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.52e8T + 1.71e21T^{2} \) |
| 47 | \( 1 + 7.15e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 4.29e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 2.55e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 8.23e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.12e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 9.39e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.92e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.72e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.22e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.68e11T + 6.73e25T^{2} \) |
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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
−10.07098325650781937855476550611, −8.874372494471711230615045847164, −7.23187377125351342616358906307, −6.48337385289413333473856980565, −5.82826118543464741395533318831, −4.34856107173671820714028089097, −3.53296359848193629620948006827, −3.34259721727908366452511829949, −1.15883885365204213649940736855, 0,
1.15883885365204213649940736855, 3.34259721727908366452511829949, 3.53296359848193629620948006827, 4.34856107173671820714028089097, 5.82826118543464741395533318831, 6.48337385289413333473856980565, 7.23187377125351342616358906307, 8.874372494471711230615045847164, 10.07098325650781937855476550611
