| L(s) = 1 | − 7.20·2-s − 12.4·3-s − 12.1·4-s + 89.5·6-s − 621.·7-s + 548.·8-s − 574.·9-s + 150.·12-s + 4.47e3·14-s − 3.17e3·16-s + 9.18e3·17-s + 4.13e3·18-s + 7.72e3·21-s − 6.81e3·24-s + 1.56e4·25-s + 1.62e4·27-s + 7.52e3·28-s − 1.22e4·32-s − 6.61e4·34-s + 6.95e3·36-s − 7.79e4·37-s − 5.56e4·42-s − 1.03e5·47-s + 3.94e4·48-s + 2.68e5·49-s − 1.12e5·50-s − 1.14e5·51-s + ⋯ |
| L(s) = 1 | − 0.900·2-s − 0.460·3-s − 0.189·4-s + 0.414·6-s − 1.81·7-s + 1.07·8-s − 0.788·9-s + 0.0871·12-s + 1.63·14-s − 0.774·16-s + 1.86·17-s + 0.709·18-s + 0.834·21-s − 0.492·24-s + 25-s + 0.823·27-s + 0.342·28-s − 0.373·32-s − 1.68·34-s + 0.149·36-s − 1.53·37-s − 0.751·42-s − 47-s + 0.356·48-s + 2.28·49-s − 0.900·50-s − 0.860·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4655694466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4655694466\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 + 1.03e5T \) |
| good | 2 | \( 1 + 7.20T + 64T^{2} \) |
| 3 | \( 1 + 12.4T + 729T^{2} \) |
| 5 | \( 1 - 1.56e4T^{2} \) |
| 7 | \( 1 + 621.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.77e6T^{2} \) |
| 13 | \( 1 - 4.82e6T^{2} \) |
| 17 | \( 1 - 9.18e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.48e8T^{2} \) |
| 29 | \( 1 - 5.94e8T^{2} \) |
| 31 | \( 1 - 8.87e8T^{2} \) |
| 37 | \( 1 + 7.79e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.32e9T^{2} \) |
| 53 | \( 1 - 5.19e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.66e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 4.49e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 9.04e10T^{2} \) |
| 71 | \( 1 + 6.84e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.51e11T^{2} \) |
| 79 | \( 1 - 7.59e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 4.44e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 7.19e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.68e6T + 8.32e11T^{2} \) |
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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
−14.42526523071268920827974954765, −13.17170970127952026400794694313, −12.09667476702416647234550267549, −10.46875621729857729612032385012, −9.677575655817845374607271434710, −8.520845052144336765644002352199, −6.94215128416530235568659429805, −5.50484771191838342736347785236, −3.31334675300618411328142513539, −0.61764543597343123090841652923,
0.61764543597343123090841652923, 3.31334675300618411328142513539, 5.50484771191838342736347785236, 6.94215128416530235568659429805, 8.520845052144336765644002352199, 9.677575655817845374607271434710, 10.46875621729857729612032385012, 12.09667476702416647234550267549, 13.17170970127952026400794694313, 14.42526523071268920827974954765
