L(s) = 1 | + (−5.55 + 1.04i)2-s + (9.77 − 9.77i)3-s + (29.7 − 11.6i)4-s + (17.6 + 17.6i)5-s + (−44.0 + 64.5i)6-s + 103. i·7-s + (−153. + 96.0i)8-s + 51.9i·9-s + (−116. − 79.7i)10-s + (1.18 + 1.18i)11-s + (177. − 405. i)12-s + (−196. + 196. i)13-s + (−108. − 576. i)14-s + 345.·15-s + (752. − 695. i)16-s + 274.·17-s + ⋯ |
L(s) = 1 | + (−0.982 + 0.185i)2-s + (0.626 − 0.626i)3-s + (0.931 − 0.364i)4-s + (0.316 + 0.316i)5-s + (−0.499 + 0.732i)6-s + 0.799i·7-s + (−0.847 + 0.530i)8-s + 0.213i·9-s + (−0.369 − 0.252i)10-s + (0.00294 + 0.00294i)11-s + (0.355 − 0.812i)12-s + (−0.322 + 0.322i)13-s + (−0.148 − 0.785i)14-s + 0.396·15-s + (0.734 − 0.678i)16-s + 0.230·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.11100 + 0.711136i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11100 + 0.711136i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.55 - 1.04i)T \) |
| 5 | \( 1 + (-17.6 - 17.6i)T \) |
good | 3 | \( 1 + (-9.77 + 9.77i)T - 243iT^{2} \) |
| 7 | \( 1 - 103. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-1.18 - 1.18i)T + 1.61e5iT^{2} \) |
| 13 | \( 1 + (196. - 196. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 - 274.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (464. - 464. i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 - 3.17e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (3.61e3 - 3.61e3i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 - 8.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-5.38e3 - 5.38e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.27e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-2.05e3 - 2.05e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + 19.2T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-2.84e3 - 2.84e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (1.60e4 + 1.60e4i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (-2.82e4 + 2.82e4i)T - 8.44e8iT^{2} \) |
| 67 | \( 1 + (4.16e4 - 4.16e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.53e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 2.47e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.18e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + (7.37e4 - 7.37e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 3.10e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 4.83e4T + 8.58e9T^{2} \) |
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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
−13.76308042333599549908654253291, −12.42272016256137751981420201790, −11.30694126260605696343759490862, −10.01659361837313948332406221175, −8.963106193680704346173924872365, −7.962916103425261252962505387177, −6.93001151031587881245994430582, −5.59024709818011434079421671582, −2.77615289984208802932849702498, −1.65331777810643451632619566351,
0.72588590877140002971982468468, 2.69119318949375170322656647631, 4.22936530193099722574561355178, 6.36112135806175542114586947839, 7.78175908458725618234287242100, 8.851101357911151366420240616691, 9.829725534811597094625469092616, 10.54406262440389400915633604049, 11.93865386749348658380412432686, 13.16528426640587887858820249535