L(s) = 1 | + (−1.33 + 0.459i)2-s + (1.57 − 1.22i)4-s + (2.14 − 0.779i)5-s + (−3.55 + 2.04i)7-s + (−1.54 + 2.36i)8-s + (−2.50 + 2.02i)10-s + (−3.61 − 2.08i)11-s + (−0.374 − 0.446i)13-s + (3.80 − 4.37i)14-s + (0.982 − 3.87i)16-s + (0.573 − 3.25i)17-s + (−0.458 − 4.33i)19-s + (2.42 − 3.85i)20-s + (5.80 + 1.13i)22-s + (0.862 − 2.37i)23-s + ⋯ |
L(s) = 1 | + (−0.945 + 0.324i)2-s + (0.789 − 0.614i)4-s + (0.957 − 0.348i)5-s + (−1.34 + 0.774i)7-s + (−0.547 + 0.837i)8-s + (−0.792 + 0.640i)10-s + (−1.09 − 0.630i)11-s + (−0.103 − 0.123i)13-s + (1.01 − 1.16i)14-s + (0.245 − 0.969i)16-s + (0.138 − 0.788i)17-s + (−0.105 − 0.994i)19-s + (0.541 − 0.862i)20-s + (1.23 + 0.241i)22-s + (0.179 − 0.494i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.217856 - 0.358323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.217856 - 0.358323i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.33 - 0.459i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.458 + 4.33i)T \) |
good | 5 | \( 1 + (-2.14 + 0.779i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (3.55 - 2.04i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.61 + 2.08i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.374 + 0.446i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.573 + 3.25i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.862 + 2.37i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (8.34 - 1.47i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.386 + 0.670i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.23iT - 37T^{2} \) |
| 41 | \( 1 + (-4.51 + 5.37i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.55 + 4.27i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (4.84 - 0.854i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.232 + 0.639i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.368 + 2.08i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.91 + 1.05i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.44 - 13.8i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-12.4 + 4.51i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (8.59 + 7.20i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (7.91 + 6.63i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.29 - 0.747i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.52 + 11.3i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 1.78i)T + (91.1 + 33.1i)T^{2} \) |
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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
−9.947378239164452297191600006312, −9.239816400534490106624847725130, −8.832102336730941663921050219599, −7.60025570774952148157231928625, −6.67241420364978986217374355356, −5.73963561279344953152159235817, −5.28749423179571454724213289273, −3.04653180563394141472932056150, −2.22971136159202489411721441634, −0.27462775792372801053390827594,
1.73568225554566499438996567216, 2.88319784369919886184005556204, 3.91103347697617397771308540982, 5.72382642059063538734845320515, 6.48212288475332197348521361685, 7.35512521535977535443916473297, 8.132509433932349553509732718029, 9.545772143804947233665253307298, 9.822836886221967285655488497734, 10.42416518565966859101400997466