L(s) = 1 | + (1.03 − 0.961i)2-s + (−2.66 + 1.54i)3-s + (0.149 − 1.99i)4-s + (0.5 − 0.866i)5-s + (−1.28 + 4.16i)6-s + (−2.53 + 0.754i)7-s + (−1.76 − 2.21i)8-s + (3.24 − 5.61i)9-s + (−0.314 − 1.37i)10-s + (−1.64 − 2.84i)11-s + (2.67 + 5.55i)12-s − 6.72·13-s + (−1.90 + 3.22i)14-s + 3.08i·15-s + (−3.95 − 0.595i)16-s + (3.32 − 1.91i)17-s + ⋯ |
L(s) = 1 | + (0.733 − 0.680i)2-s + (−1.54 + 0.889i)3-s + (0.0746 − 0.997i)4-s + (0.223 − 0.387i)5-s + (−0.524 + 1.69i)6-s + (−0.958 + 0.285i)7-s + (−0.623 − 0.781i)8-s + (1.08 − 1.87i)9-s + (−0.0995 − 0.435i)10-s + (−0.495 − 0.857i)11-s + (0.771 + 1.60i)12-s − 1.86·13-s + (−0.508 + 0.860i)14-s + 0.795i·15-s + (−0.988 − 0.148i)16-s + (0.806 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0768393 - 0.488445i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0768393 - 0.488445i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.03 + 0.961i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.53 - 0.754i)T \) |
good | 3 | \( 1 + (2.66 - 1.54i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.64 + 2.84i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.72T + 13T^{2} \) |
| 17 | \( 1 + (-3.32 + 1.91i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.618 + 0.357i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.09 + 0.631i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.04iT - 29T^{2} \) |
| 31 | \( 1 + (0.0335 + 0.0581i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.498 - 0.287i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.230iT - 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + (-4.23 + 7.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.16 - 1.25i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.986 + 0.569i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.0888 + 0.153i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.92 + 12.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.0iT - 71T^{2} \) |
| 73 | \( 1 + (3.89 - 2.25i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.66 + 5.58i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.24iT - 83T^{2} \) |
| 89 | \( 1 + (13.3 + 7.68i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.76iT - 97T^{2} \) |
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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
−11.56782745766841174698174990245, −10.51411544219968522713549037518, −9.941873852643475851486262623197, −9.229654368625420761602939971055, −7.00861732677098126294384089669, −5.80090936834385774460564133334, −5.35649282153875178606630466951, −4.33876202348124406561286737008, −2.92659300722153648664985228775, −0.33227754752332076960505813890,
2.50571879838100401361672478933, 4.44778215589710446196722743885, 5.52714262470188427465391413521, 6.25764777793592863145904053071, 7.29725178626296755026189283286, 7.54536130639853060933186020952, 9.713242606040028503125314357374, 10.54461197477493474691039236575, 11.78749295911419829221036832817, 12.54951945387528753513863410747