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
−13.99524517399761332637881765514, −12.67895497729706063441700220867, −12.25337225529887578170373639309, −10.76902491094039043256439157132, −9.560628838996668686128169895108, −7.56312137038200481961178038345, −6.87805361424906816530500756329, −5.30911214508575554893580126402, −4.44555496882041208879018615646, −2.78239060434164072379059359803,
2.27751616867130468029526039840, 3.85372150776860430968747143008, 5.17376963098120793189661771929, 6.07506819984446625212410755106, 7.85186253270525682366054258793, 9.336057973489378908101722945502, 10.76890973521706128214209526873, 11.84977623309730268423010468779, 12.45843767796916600334242766780, 13.48874310397411749696033646524