L(s) = 1 | + (2.96 − 0.963i)2-s + (4.62 − 3.35i)4-s + (0.439 − 1.35i)5-s + (2.23 + 3.08i)7-s + (3.13 − 4.32i)8-s − 4.43i·10-s + (0.450 − 10.9i)11-s + (−20.6 + 6.69i)13-s + (9.60 + 6.97i)14-s + (−1.91 + 5.90i)16-s + (16.1 + 5.24i)17-s + (−11.4 + 15.8i)19-s + (−2.50 − 7.72i)20-s + (−9.25 − 33.0i)22-s − 4.82·23-s + ⋯ |
L(s) = 1 | + (1.48 − 0.481i)2-s + (1.15 − 0.839i)4-s + (0.0878 − 0.270i)5-s + (0.319 + 0.440i)7-s + (0.392 − 0.540i)8-s − 0.443i·10-s + (0.0409 − 0.999i)11-s + (−1.58 + 0.515i)13-s + (0.685 + 0.498i)14-s + (−0.119 + 0.369i)16-s + (0.950 + 0.308i)17-s + (−0.604 + 0.831i)19-s + (−0.125 − 0.386i)20-s + (−0.420 − 1.50i)22-s − 0.209·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.55146 - 0.855843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.55146 - 0.855843i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-0.450 + 10.9i)T \) |
good | 2 | \( 1 + (-2.96 + 0.963i)T + (3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (-0.439 + 1.35i)T + (-20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-2.23 - 3.08i)T + (-15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (20.6 - 6.69i)T + (136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (-16.1 - 5.24i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (11.4 - 15.8i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + 4.82T + 529T^{2} \) |
| 29 | \( 1 + (26.7 + 36.7i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (6.47 + 19.9i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-17.4 + 12.6i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-21.5 + 29.6i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 10.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-53.7 - 39.0i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (25.4 + 78.4i)T + (-2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (51.7 - 37.6i)T + (1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 3.37i)T + (3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 22.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-13.7 + 42.2i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (41.1 + 56.6i)T + (-1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-132. + 43.1i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-106. - 34.6i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 + 63.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-31.2 - 96.1i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−13.48874310397411749696033646524, −12.45843767796916600334242766780, −11.84977623309730268423010468779, −10.76890973521706128214209526873, −9.336057973489378908101722945502, −7.85186253270525682366054258793, −6.07506819984446625212410755106, −5.17376963098120793189661771929, −3.85372150776860430968747143008, −2.27751616867130468029526039840,
2.78239060434164072379059359803, 4.44555496882041208879018615646, 5.30911214508575554893580126402, 6.87805361424906816530500756329, 7.56312137038200481961178038345, 9.560628838996668686128169895108, 10.76902491094039043256439157132, 12.25337225529887578170373639309, 12.67895497729706063441700220867, 13.99524517399761332637881765514