| L(s) = 1 | + (2.26 − 12.8i)2-s + (−39.1 − 14.2i)4-s + (4.19 − 3.51i)5-s + (−1.35e3 + 491. i)7-s + (562. − 973. i)8-s + (−35.6 − 61.7i)10-s + (916. + 768. i)11-s + (294. + 1.66e3i)13-s + (3.25e3 + 1.84e4i)14-s + (−1.53e4 − 1.28e4i)16-s + (1.09e4 + 1.90e4i)17-s + (−1.67e4 + 2.90e4i)19-s + (−214. + 77.9i)20-s + (1.19e4 − 1.00e4i)22-s + (1.76e4 + 6.41e3i)23-s + ⋯ |
| L(s) = 1 | + (0.199 − 1.13i)2-s + (−0.305 − 0.111i)4-s + (0.0149 − 0.0125i)5-s + (−1.48 + 0.541i)7-s + (0.388 − 0.672i)8-s + (−0.0112 − 0.0195i)10-s + (0.207 + 0.174i)11-s + (0.0371 + 0.210i)13-s + (0.316 + 1.79i)14-s + (−0.934 − 0.783i)16-s + (0.542 + 0.940i)17-s + (−0.560 + 0.969i)19-s + (−0.00598 + 0.00217i)20-s + (0.238 − 0.200i)22-s + (0.302 + 0.110i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.23968 + 0.358737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23968 + 0.358737i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-2.26 + 12.8i)T + (-120. - 43.7i)T^{2} \) |
| 5 | \( 1 + (-4.19 + 3.51i)T + (1.35e4 - 7.69e4i)T^{2} \) |
| 7 | \( 1 + (1.35e3 - 491. i)T + (6.30e5 - 5.29e5i)T^{2} \) |
| 11 | \( 1 + (-916. - 768. i)T + (3.38e6 + 1.91e7i)T^{2} \) |
| 13 | \( 1 + (-294. - 1.66e3i)T + (-5.89e7 + 2.14e7i)T^{2} \) |
| 17 | \( 1 + (-1.09e4 - 1.90e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.67e4 - 2.90e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.76e4 - 6.41e3i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (3.07e4 - 1.74e5i)T + (-1.62e10 - 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-1.99e5 - 7.26e4i)T + (2.10e10 + 1.76e10i)T^{2} \) |
| 37 | \( 1 + (1.17e5 + 2.02e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (9.52e4 + 5.40e5i)T + (-1.83e11 + 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-3.79e5 - 3.18e5i)T + (4.72e10 + 2.67e11i)T^{2} \) |
| 47 | \( 1 + (-3.02e4 + 1.09e4i)T + (3.88e11 - 3.25e11i)T^{2} \) |
| 53 | \( 1 - 1.06e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (2.28e6 - 1.91e6i)T + (4.32e11 - 2.45e12i)T^{2} \) |
| 61 | \( 1 + (2.42e6 - 8.81e5i)T + (2.40e12 - 2.02e12i)T^{2} \) |
| 67 | \( 1 + (1.72e5 + 9.76e5i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (3.73e5 + 6.46e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.19e6 - 3.80e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (6.31e5 - 3.57e6i)T + (-1.80e13 - 6.56e12i)T^{2} \) |
| 83 | \( 1 + (8.94e5 - 5.07e6i)T + (-2.54e13 - 9.28e12i)T^{2} \) |
| 89 | \( 1 + (-2.30e6 + 3.98e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.24e7 + 1.04e7i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−12.51899146041632390655644733968, −12.31006576804789940118886049830, −10.80360618738131676202277664876, −9.956272480427031614397194638810, −8.931994323661069159713121891800, −7.10099205603331881597708068486, −5.88303064324948873085885501273, −3.91011094081189206686727767534, −2.94853500961300248866200184789, −1.48185122156918558297552974392,
0.39732928360355974352283734731, 2.82514617066214485206488448540, 4.52303636905848477416373074508, 6.09058844448221409216218263542, 6.77531242884191918648469308317, 7.920165861846945934893770404557, 9.355453029311494523870075818994, 10.46630910163077185094542347421, 11.85444404490738442868788204130, 13.28877386999614055654936549900
