| L(s) = 1 | + (16.1 + 28.0i)2-s + (−266. + 461. i)4-s + (−1.03e3 + 1.79e3i)5-s + (5.25e3 + 9.10e3i)7-s − 692.·8-s − 6.69e4·10-s + (1.58e4 + 2.73e4i)11-s + (−2.38e4 + 4.12e4i)13-s + (−1.70e5 + 2.94e5i)14-s + (1.25e5 + 2.17e5i)16-s + 3.44e4·17-s + 7.63e5·19-s + (−5.52e5 − 9.57e5i)20-s + (−5.11e5 + 8.85e5i)22-s + (−7.42e5 + 1.28e6i)23-s + ⋯ |
| L(s) = 1 | + (0.714 + 1.23i)2-s + (−0.520 + 0.902i)4-s + (−0.741 + 1.28i)5-s + (0.827 + 1.43i)7-s − 0.0598·8-s − 2.11·10-s + (0.325 + 0.563i)11-s + (−0.231 + 0.400i)13-s + (−1.18 + 2.04i)14-s + (0.478 + 0.828i)16-s + 0.100·17-s + 1.34·19-s + (−0.772 − 1.33i)20-s + (−0.465 + 0.805i)22-s + (−0.553 + 0.958i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.01291 - 2.78296i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01291 - 2.78296i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-16.1 - 28.0i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (1.03e3 - 1.79e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-5.25e3 - 9.10e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-1.58e4 - 2.73e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (2.38e4 - 4.12e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 - 3.44e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.63e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (7.42e5 - 1.28e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (2.40e6 + 4.16e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-3.91e6 + 6.77e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 9.43e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (6.14e6 - 1.06e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.87e7 + 3.24e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (9.91e6 + 1.71e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 - 5.78e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-1.42e7 + 2.46e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (2.22e7 + 3.85e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-9.09e7 + 1.57e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.53e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.55e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-1.63e8 - 2.82e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (2.50e7 + 4.34e7i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + 1.59e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-7.46e7 - 1.29e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−13.62738083325262096955730244238, −11.92181778047236727216209875997, −11.44304180467262353151070702896, −9.708478925893562504989831541745, −8.067523241772098464738356000678, −7.35703911463069152312683743422, −6.19504326526645528409701992984, −5.13212776663107676032529576462, −3.76770847262583606637455834903, −2.16323991930885370903791831891,
0.73866229231119596477139201001, 1.31236360864619826941037793309, 3.34522235015321195888823863944, 4.36135092702966919326975063530, 5.09383226598210366731435037352, 7.43471581245596826377457810162, 8.430351748463460878215504806200, 10.02331482086412284946468401848, 11.07241437729438507488890589638, 11.86350274353586018036654634743
