L(s) = 1 | − 3.09·2-s + 9·3-s − 22.4·4-s + 13.7·5-s − 27.8·6-s + 168.·8-s + 81·9-s − 42.6·10-s − 3.66·11-s − 201.·12-s − 780.·13-s + 124.·15-s + 197.·16-s + 50.0·17-s − 250.·18-s + 1.06e3·19-s − 309.·20-s + 11.3·22-s + 4.10e3·23-s + 1.51e3·24-s − 2.93e3·25-s + 2.41e3·26-s + 729·27-s − 1.48e3·29-s − 383.·30-s + 5.51e3·31-s − 5.99e3·32-s + ⋯ |
L(s) = 1 | − 0.546·2-s + 0.577·3-s − 0.701·4-s + 0.246·5-s − 0.315·6-s + 0.929·8-s + 0.333·9-s − 0.134·10-s − 0.00913·11-s − 0.404·12-s − 1.28·13-s + 0.142·15-s + 0.193·16-s + 0.0420·17-s − 0.182·18-s + 0.675·19-s − 0.173·20-s + 0.00499·22-s + 1.61·23-s + 0.536·24-s − 0.939·25-s + 0.699·26-s + 0.192·27-s − 0.328·29-s − 0.0778·30-s + 1.03·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.477236867\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.477236867\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 3.09T + 32T^{2} \) |
| 5 | \( 1 - 13.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 3.66T + 1.61e5T^{2} \) |
| 13 | \( 1 + 780.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 50.0T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.14e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.76e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.94e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.92e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.61e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.89e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.77e4T + 8.58e9T^{2} \) |
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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
−12.30028561429513426527789654474, −10.85672909798467817830313467247, −9.678513540867912274514677962984, −9.273393326089731921418152696762, −8.017601503321021590295771659388, −7.19615050535949132987662130688, −5.38461592096716951710077212518, −4.21513151076043861357561769065, −2.58125932033867867763265398515, −0.876658401299134050964387616418,
0.876658401299134050964387616418, 2.58125932033867867763265398515, 4.21513151076043861357561769065, 5.38461592096716951710077212518, 7.19615050535949132987662130688, 8.017601503321021590295771659388, 9.273393326089731921418152696762, 9.678513540867912274514677962984, 10.85672909798467817830313467247, 12.30028561429513426527789654474