| L(s) = 1 | + 9i·3-s − 16i·7-s − 81·9-s − 564·11-s + 370i·13-s − 1.08e3i·17-s + 2.86e3·19-s + 144·21-s − 1.58e3i·23-s − 729i·27-s − 1.13e3·29-s − 6.01e3·31-s − 5.07e3i·33-s − 538i·37-s − 3.33e3·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.123i·7-s − 0.333·9-s − 1.40·11-s + 0.607i·13-s − 0.911i·17-s + 1.81·19-s + 0.0712·21-s − 0.624i·23-s − 0.192i·27-s − 0.250·29-s − 1.12·31-s − 0.811i·33-s − 0.0646i·37-s − 0.350·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.551735873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551735873\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 16iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 564T + 1.61e5T^{2} \) |
| 13 | \( 1 - 370iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.08e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.86e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.58e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 538iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.13e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.44e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.02e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.47e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.42e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.11e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.26e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.95e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.40e5iT - 8.58e9T^{2} \) |
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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
−10.82496157060521223329901050987, −9.849607469096474045812989396319, −9.141965045630881007810171950649, −7.907091319559912372276657153289, −7.08055740612991876564117542543, −5.57422256013872208556669576378, −4.86992263637292555453780463350, −3.51841792855409096749709270200, −2.37418720420852210045740984550, −0.51624678140869409620760055258,
0.942628887434728886593461104622, 2.37642269413640562830914766148, 3.50902051849129767067991588583, 5.22622089462969165116924676058, 5.86071725854605011670080534104, 7.38993262570852929343217259988, 7.83178075360141745277805124108, 9.010331075890822638387287322738, 10.12389773340688517323163546642, 10.97110092408983469465050690170
