L(s) = 1 | + (−0.762 − 1.55i)3-s + (0.705 − 1.22i)5-s + (1.17 + 2.02i)7-s + (−1.83 + 2.37i)9-s + (1.30 + 2.25i)11-s + (1.26 − 2.18i)13-s + (−2.43 − 0.165i)15-s + 4.94·17-s + 1.00·19-s + (2.26 − 3.36i)21-s + (−1.50 + 2.60i)23-s + (1.50 + 2.60i)25-s + (5.08 + 1.05i)27-s + (−0.0708 − 0.122i)29-s + (−4.77 + 8.26i)31-s + ⋯ |
L(s) = 1 | + (−0.440 − 0.897i)3-s + (0.315 − 0.546i)5-s + (0.442 + 0.766i)7-s + (−0.612 + 0.790i)9-s + (0.392 + 0.679i)11-s + (0.350 − 0.606i)13-s + (−0.629 − 0.0427i)15-s + 1.19·17-s + 0.231·19-s + (0.493 − 0.734i)21-s + (−0.314 + 0.544i)23-s + (0.300 + 0.521i)25-s + (0.979 + 0.202i)27-s + (−0.0131 − 0.0228i)29-s + (−0.856 + 1.48i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.635755181\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.635755181\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.762 + 1.55i)T \) |
good | 5 | \( 1 + (-0.705 + 1.22i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.17 - 2.02i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.30 - 2.25i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.26 + 2.18i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.94T + 17T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 + (1.50 - 2.60i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0708 + 0.122i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.77 - 8.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.00T + 37T^{2} \) |
| 41 | \( 1 + (-4.33 + 7.50i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.15 - 5.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.24 + 5.62i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6.02T + 53T^{2} \) |
| 59 | \( 1 + (-5.64 + 9.77i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.45 + 5.99i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.154 - 0.268i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.24T + 71T^{2} \) |
| 73 | \( 1 + 6.78T + 73T^{2} \) |
| 79 | \( 1 + (4.99 + 8.65i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.47 + 6.01i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + (-7.44 - 12.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.616798388388190806047472163658, −8.861695304098726656588478068377, −7.977285399981125173854127503642, −7.36574939175786519700096650930, −6.26355325310889949766912308806, −5.46926978161993148178188681511, −4.97010355658943709524620522516, −3.37341071456529645510869600100, −2.03527863658142283191770112420, −1.13468720726333532403441617160,
0.989769989168480871328111870616, 2.77911347462985339400941677536, 3.88010309172953264633220023923, 4.49524159721015294784357152450, 5.78828163382819729204665426626, 6.23559355955967712693417558612, 7.37170787711403356293204845394, 8.261448997118398880005471375091, 9.304927422740215997169066518717, 9.900209381585410809445636911234