| L(s) = 1 | + (18.1 + 3.20i)2-s + (79.1 + 28.8i)4-s + (−462. − 551. i)5-s + (1.64e3 − 597. i)7-s + (−2.74e3 − 1.58e3i)8-s + (−6.63e3 − 1.15e4i)10-s + (−9.01e3 + 1.07e4i)11-s + (9.04e3 + 5.12e4i)13-s + (3.17e4 − 5.59e3i)14-s + (−6.12e4 − 5.14e4i)16-s + (4.97e4 − 2.86e4i)17-s + (−1.16e5 + 2.02e5i)19-s + (−2.07e4 − 5.69e4i)20-s + (−1.98e5 + 1.66e5i)22-s + (−7.95e4 + 2.18e5i)23-s + ⋯ |
| L(s) = 1 | + (1.13 + 0.200i)2-s + (0.309 + 0.112i)4-s + (−0.740 − 0.882i)5-s + (0.684 − 0.249i)7-s + (−0.669 − 0.386i)8-s + (−0.663 − 1.15i)10-s + (−0.615 + 0.733i)11-s + (0.316 + 1.79i)13-s + (0.826 − 0.145i)14-s + (−0.935 − 0.784i)16-s + (0.595 − 0.343i)17-s + (−0.896 + 1.55i)19-s + (−0.129 − 0.356i)20-s + (−0.846 + 0.710i)22-s + (−0.284 + 0.781i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.378535 + 0.921239i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378535 + 0.921239i\) |
| \(L(5)\) |
|
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 | \( 1 + (-18.1 - 3.20i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (462. + 551. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (-1.64e3 + 597. i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (9.01e3 - 1.07e4i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (-9.04e3 - 5.12e4i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (-4.97e4 + 2.86e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.16e5 - 2.02e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (7.95e4 - 2.18e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (1.47e5 + 2.60e4i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (1.38e6 + 5.04e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (-6.97e5 - 1.20e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-1.41e6 + 2.48e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (6.94e4 + 5.82e4i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (2.27e6 + 6.26e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 - 3.96e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-3.41e6 - 4.07e6i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (-1.58e6 + 5.75e5i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (-1.88e6 - 1.06e7i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (3.79e7 - 2.19e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (9.39e6 - 1.62e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.71e5 + 9.74e5i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (2.50e7 + 4.41e6i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (-3.33e7 - 1.92e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (1.11e8 + 9.33e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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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
−13.10635985820518291561929867900, −12.21524990552503050837408393273, −11.46528130151849978146268023856, −9.730248141193209638728961314822, −8.468288594976633514778708089139, −7.26413848303201406006240486152, −5.69190550700159938466062100216, −4.51323834643289901255289656011, −3.91859151867185037772099538644, −1.69401190585144433687068520292,
0.19913480075292314794631428415, 2.67438877008604153191073207302, 3.52796857509180597570933790820, 4.96283448215756741414108644009, 6.02438043272718824471666442942, 7.67524274442876055140236782660, 8.648087842371858800498600386244, 10.77667169525906896054221535207, 11.15164222610649186058096675512, 12.54639326019291535965225835678
