L(s) = 1 | + (1.49 + 3.70i)2-s + 6.26i·3-s + (−11.5 + 11.1i)4-s + 49.7·5-s + (−23.2 + 9.39i)6-s + 51.4i·7-s + (−58.4 − 26.0i)8-s + 41.6·9-s + (74.5 + 184. i)10-s − 67.5i·11-s + (−69.6 − 72.1i)12-s − 124.·13-s + (−190. + 77.1i)14-s + 311. i·15-s + (8.83 − 255. i)16-s − 274.·17-s + ⋯ |
L(s) = 1 | + (0.374 + 0.927i)2-s + 0.696i·3-s + (−0.719 + 0.694i)4-s + 1.99·5-s + (−0.645 + 0.261i)6-s + 1.05i·7-s + (−0.913 − 0.406i)8-s + 0.514·9-s + (0.745 + 1.84i)10-s − 0.557i·11-s + (−0.484 − 0.501i)12-s − 0.739·13-s + (−0.974 + 0.393i)14-s + 1.38i·15-s + (0.0345 − 0.999i)16-s − 0.950·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.943158 + 2.22253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.943158 + 2.22253i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.49 - 3.70i)T \) |
| 19 | \( 1 - 82.8iT \) |
good | 3 | \( 1 - 6.26iT - 81T^{2} \) |
| 5 | \( 1 - 49.7T + 625T^{2} \) |
| 7 | \( 1 - 51.4iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 67.5iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 124.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 274.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 914. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 414.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 823. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.91e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 457.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.52e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 742. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.33e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 2.25e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.11e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.02e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 8.13e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 578.T + 2.83e7T^{2} \) |
| 79 | \( 1 + 373. iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 548. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.82e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 4.83e3T + 8.85e7T^{2} \) |
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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
−14.36288630240301826961833160277, −13.27907420254542659630149420930, −12.47282733325323660200949024258, −10.48211009155093148662366451081, −9.414751814351762654353138909837, −8.760304483159997136480557259971, −6.68503148473613897527638712269, −5.71860082715531259611251060804, −4.73510480352172181945050738573, −2.51563838756778938675209629784,
1.28430133050614145339040122929, 2.32847098502824241523128422803, 4.55554894401694927398995657562, 5.98723946361149330256650032436, 7.23770963868883609463784086128, 9.464052659381407195081733474038, 9.913687314444618812518780114559, 11.08325533177863098306021391341, 12.67688933168330164655928219373, 13.36486833062611606447056292199