| L(s) = 1 | + (−26.0 − 15.0i)2-s + (324. + 562. i)4-s + (−189. + 109. i)5-s + (1.40e3 − 2.43e3i)7-s − 1.18e4i·8-s + 6.57e3·10-s + (4.50e3 + 2.59e3i)11-s + (−5.26e3 − 9.11e3i)13-s + (−7.32e4 + 4.22e4i)14-s + (−9.49e4 + 1.64e5i)16-s + 1.52e4i·17-s − 9.26e4·19-s + (−1.22e5 − 7.09e4i)20-s + (−7.82e4 − 1.35e5i)22-s + (5.31e4 − 3.06e4i)23-s + ⋯ |
| L(s) = 1 | + (−1.62 − 0.940i)2-s + (1.26 + 2.19i)4-s + (−0.302 + 0.174i)5-s + (0.585 − 1.01i)7-s − 2.88i·8-s + 0.657·10-s + (0.307 + 0.177i)11-s + (−0.184 − 0.319i)13-s + (−1.90 + 1.10i)14-s + (−1.44 + 2.50i)16-s + 0.182i·17-s − 0.711·19-s + (−0.767 − 0.443i)20-s + (−0.333 − 0.578i)22-s + (0.189 − 0.109i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0506764 + 0.244191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0506764 + 0.244191i\) |
| \(L(5)\) |
|
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 | \( 1 + (26.0 + 15.0i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (189. - 109. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-1.40e3 + 2.43e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-4.50e3 - 2.59e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (5.26e3 + 9.11e3i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 - 1.52e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 9.26e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-5.31e4 + 3.06e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (7.59e5 + 4.38e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (8.14e5 + 1.41e6i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + 1.66e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (4.15e6 - 2.39e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-2.56e5 + 4.44e5i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (1.08e6 + 6.25e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + 8.87e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-8.34e6 + 4.81e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.51e6 - 7.81e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (5.32e6 + 9.22e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.16e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.00e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (2.83e7 - 4.90e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-1.29e6 - 7.45e5i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 + 5.42e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (3.06e6 - 5.31e6i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−15.08315741053493800633879441498, −13.11042144880178819079050619589, −11.61611936910323528196919513306, −10.78318211489154624960667555465, −9.644045024716554060242183749912, −8.167029483215785512785108797296, −7.17703111198431697599328043676, −3.76399549508461902538024027846, −1.78009765520547455941052454132, −0.18969101655857103297993495875,
1.76359956991257508703764774753, 5.42841516787625876081415239952, 6.95060447140076511779302009452, 8.382203815581694561881378475335, 9.125319624685386792911829966502, 10.68073450842387269731543703588, 11.97188581496206723045027343951, 14.41568589520676684892240917699, 15.35654080165362219402344082051, 16.34000061354843194638947267600
