| L(s) = 1 | + (−1.23 − 3.80i)2-s + 28.4·3-s + (−12.9 + 9.40i)4-s + (−21.6 + 15.7i)5-s + (−35.1 − 108. i)6-s + (−76.2 + 234. i)7-s + (51.7 + 37.6i)8-s + 566.·9-s + (86.6 + 62.9i)10-s + (80.1 + 58.2i)11-s + (−368. + 267. i)12-s + (293. + 903. i)13-s + 986.·14-s + (−616. + 447. i)15-s + (79.1 − 243. i)16-s + (−914. − 664. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + 1.82·3-s + (−0.404 + 0.293i)4-s + (−0.387 + 0.281i)5-s + (−0.398 − 1.22i)6-s + (−0.587 + 1.80i)7-s + (0.286 + 0.207i)8-s + 2.33·9-s + (0.274 + 0.199i)10-s + (0.199 + 0.145i)11-s + (−0.738 + 0.536i)12-s + (0.481 + 1.48i)13-s + 1.34·14-s + (−0.707 + 0.513i)15-s + (0.0772 − 0.237i)16-s + (−0.767 − 0.557i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.39775 + 0.639970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.39775 + 0.639970i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 41 | \( 1 + (7.23e3 + 7.96e3i)T \) |
| good | 3 | \( 1 - 28.4T + 243T^{2} \) |
| 5 | \( 1 + (21.6 - 15.7i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (76.2 - 234. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-80.1 - 58.2i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-293. - 903. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (914. + 664. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-277. + 855. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (587. + 1.80e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.77e3 + 3.46e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.55e3 - 1.12e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.09e4 + 7.98e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 43 | \( 1 + (-4.20e3 - 1.29e4i)T + (-1.18e8 + 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-1.03e3 - 3.19e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (3.23e3 - 2.35e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (3.06e3 + 9.44e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (8.66e3 - 2.66e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-5.21e4 + 3.78e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.25e4 - 1.64e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 - 2.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-9.89e3 + 3.04e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (5.84e4 - 4.24e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−13.47474111248680341864444310013, −12.41982080356958974770899645904, −11.39774336578778037979308378979, −9.518291197261002378311734642930, −9.129141855515447737577253775406, −8.246375677876977530695635752471, −6.71047085253651279623031982529, −4.26858266802959106462466528741, −2.89045517157807109266798519420, −2.11452815720621185655855702411,
0.981454446189770931056584831891, 3.34721981979634796144141748517, 4.24409130984239317844951195205, 6.68105822556469176694070571889, 7.85328583983307578980442626746, 8.329028283319310834598157259220, 9.753866416553286732434259952935, 10.48644723876820507846919840959, 12.88162435352997008573991675531, 13.52256236979496179194977067826
