| L(s) = 1 | + 8·2-s + 50.9·3-s + 64·4-s − 511.·5-s + 407.·6-s − 547.·7-s + 512·8-s + 404.·9-s − 4.09e3·10-s − 4.16e3·11-s + 3.25e3·12-s − 1.18e4·13-s − 4.38e3·14-s − 2.60e4·15-s + 4.09e3·16-s + 9.96e3·17-s + 3.23e3·18-s + 1.99e4·19-s − 3.27e4·20-s − 2.78e4·21-s − 3.33e4·22-s + 6.63e4·23-s + 2.60e4·24-s + 1.83e5·25-s − 9.46e4·26-s − 9.07e4·27-s − 3.50e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.08·3-s + 0.5·4-s − 1.83·5-s + 0.769·6-s − 0.603·7-s + 0.353·8-s + 0.184·9-s − 1.29·10-s − 0.943·11-s + 0.544·12-s − 1.49·13-s − 0.426·14-s − 1.99·15-s + 0.250·16-s + 0.492·17-s + 0.130·18-s + 0.667·19-s − 0.915·20-s − 0.656·21-s − 0.667·22-s + 1.13·23-s + 0.384·24-s + 2.35·25-s − 1.05·26-s − 0.887·27-s − 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 31 | \( 1 + 2.97e4T \) |
| good | 3 | \( 1 - 50.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 511.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 547.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.16e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.18e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 9.96e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.63e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.78e5T + 1.72e10T^{2} \) |
| 37 | \( 1 + 3.52e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.54e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.94e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.76e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.70e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.21e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.91e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.73e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.48e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.26e6T + 8.07e13T^{2} \) |
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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
−12.91306339145363625310131242737, −12.13749055286103239672290006593, −10.92416854319758040806299185718, −9.303449740631339970450910784278, −7.81041505523801468969228685649, −7.33579989286027509323191650367, −5.03315934586403738325576737909, −3.57497483636322785977326923113, −2.78732235215970373815243734737, 0,
2.78732235215970373815243734737, 3.57497483636322785977326923113, 5.03315934586403738325576737909, 7.33579989286027509323191650367, 7.81041505523801468969228685649, 9.303449740631339970450910784278, 10.92416854319758040806299185718, 12.13749055286103239672290006593, 12.91306339145363625310131242737
