L(s) = 1 | + 4.20i·3-s + (−1.52 − 4.76i)5-s − 3.29·7-s − 8.67·9-s + 15.0·11-s − 7.02·13-s + (20.0 − 6.39i)15-s + 15.9i·17-s − 12.8·19-s − 13.8i·21-s − 3.49·23-s + (−20.3 + 14.4i)25-s + 1.36i·27-s + 13.1i·29-s − 38.2i·31-s + ⋯ |
L(s) = 1 | + 1.40i·3-s + (−0.304 − 0.952i)5-s − 0.471·7-s − 0.963·9-s + 1.36·11-s − 0.540·13-s + (1.33 − 0.426i)15-s + 0.936i·17-s − 0.678·19-s − 0.660i·21-s − 0.151·23-s + (−0.814 + 0.579i)25-s + 0.0505i·27-s + 0.453i·29-s − 1.23i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1639182454\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1639182454\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.52 + 4.76i)T \) |
good | 3 | \( 1 - 4.20iT - 9T^{2} \) |
| 7 | \( 1 + 3.29T + 49T^{2} \) |
| 11 | \( 1 - 15.0T + 121T^{2} \) |
| 13 | \( 1 + 7.02T + 169T^{2} \) |
| 17 | \( 1 - 15.9iT - 289T^{2} \) |
| 19 | \( 1 + 12.8T + 361T^{2} \) |
| 23 | \( 1 + 3.49T + 529T^{2} \) |
| 29 | \( 1 - 13.1iT - 841T^{2} \) |
| 31 | \( 1 + 38.2iT - 961T^{2} \) |
| 37 | \( 1 + 68.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 51.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 34.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 79.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 114.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 115. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 63.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 2.16iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 78.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 61.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 6.04T + 7.92e3T^{2} \) |
| 97 | \( 1 + 135. iT - 9.40e3T^{2} \) |
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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
−9.212410024121267490513765968866, −8.799705014950308302854086907097, −7.77789164163513136274621522935, −6.57719258971708812859147269851, −5.69012397089026951400008332469, −4.69991400480704162164210314627, −4.07164255763254390247416811739, −3.44201719156018194583299774009, −1.71730896227216837523701435820, −0.04770800455412122334261399308,
1.36252039550294291097103214392, 2.48479055037994093230900875552, 3.38448865662892401936509654019, 4.56288224811241228568541725168, 6.08821105854772068087430636497, 6.57117594121305847210787356709, 7.19175511123488469716772178692, 7.79991817106645327799693443218, 8.879579170769231480189529403867, 9.661380782229966425578438624495