| L(s) = 1 | + 127.·2-s + 729·3-s + 7.97e3·4-s − 1.58e4·5-s + 9.26e4·6-s + 3.38e5·7-s − 2.77e4·8-s + 5.31e5·9-s − 2.00e6·10-s − 9.34e6·11-s + 5.81e6·12-s + 1.20e6·13-s + 4.30e7·14-s − 1.15e7·15-s − 6.88e7·16-s + 8.21e7·17-s + 6.75e7·18-s − 8.01e6·19-s − 1.26e8·20-s + 2.46e8·21-s − 1.18e9·22-s + 1.37e9·23-s − 2.02e7·24-s − 9.71e8·25-s + 1.52e8·26-s + 3.87e8·27-s + 2.69e9·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.577·3-s + 0.973·4-s − 0.452·5-s + 0.811·6-s + 1.08·7-s − 0.0373·8-s + 0.333·9-s − 0.635·10-s − 1.59·11-s + 0.561·12-s + 0.0690·13-s + 1.52·14-s − 0.261·15-s − 1.02·16-s + 0.825·17-s + 0.468·18-s − 0.0390·19-s − 0.440·20-s + 0.627·21-s − 2.23·22-s + 1.93·23-s − 0.0215·24-s − 0.795·25-s + 0.0969·26-s + 0.192·27-s + 1.05·28-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
| \(L(7)\) |
\(\approx\) |
\(6.329817082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.329817082\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 - 127.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 1.58e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 3.38e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 9.34e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.20e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 8.21e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 8.01e6T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.37e9T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.06e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 7.91e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.32e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 5.29e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.43e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 4.30e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 8.06e10T + 2.60e22T^{2} \) |
| 61 | \( 1 + 2.15e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 8.36e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 6.69e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.21e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 3.05e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.99e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 8.83e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.28e12T + 6.73e25T^{2} \) |
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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
−10.71300171049261639941935517895, −9.267696773967103742092413713746, −8.003975074025205748245634366782, −7.46502901290492405648482714900, −5.86286406426699814918238698680, −4.97359532604060104239521081810, −4.24920861206519987653812314797, −3.05898247020770636106068047990, −2.35592171536863213030889524891, −0.836893784627282030117701837824,
0.836893784627282030117701837824, 2.35592171536863213030889524891, 3.05898247020770636106068047990, 4.24920861206519987653812314797, 4.97359532604060104239521081810, 5.86286406426699814918238698680, 7.46502901290492405648482714900, 8.003975074025205748245634366782, 9.267696773967103742092413713746, 10.71300171049261639941935517895
