| L(s) = 1 | + (1.33 − 0.459i)2-s + (0.220 + 0.185i)3-s + (1.57 − 1.22i)4-s + (−2.14 + 0.779i)5-s + (0.380 + 0.146i)6-s + (−3.55 + 2.04i)7-s + (1.54 − 2.36i)8-s + (−0.506 − 2.87i)9-s + (−2.50 + 2.02i)10-s + (3.61 + 2.08i)11-s + (0.575 + 0.0212i)12-s + (−0.374 − 0.446i)13-s + (−3.80 + 4.37i)14-s + (−0.616 − 0.224i)15-s + (0.982 − 3.87i)16-s + (−0.573 + 3.25i)17-s + ⋯ |
| L(s) = 1 | + (0.945 − 0.324i)2-s + (0.127 + 0.106i)3-s + (0.789 − 0.614i)4-s + (−0.957 + 0.348i)5-s + (0.155 + 0.0597i)6-s + (−1.34 + 0.774i)7-s + (0.547 − 0.837i)8-s + (−0.168 − 0.957i)9-s + (−0.792 + 0.640i)10-s + (1.09 + 0.630i)11-s + (0.166 + 0.00612i)12-s + (−0.103 − 0.123i)13-s + (−1.01 + 1.16i)14-s + (−0.159 − 0.0579i)15-s + (0.245 − 0.969i)16-s + (−0.138 + 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31048 - 0.200551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31048 - 0.200551i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 + 0.459i)T \) |
| 19 | \( 1 + (0.458 + 4.33i)T \) |
| good | 3 | \( 1 + (-0.220 - 0.185i)T + (0.520 + 2.95i)T^{2} \) |
| 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
−14.69773229466663889012370445949, −13.21897810802203606763426835136, −12.14343057734572377812375547546, −11.72180108968522663414541289463, −10.11248530020334498054816731128, −9.012987265632940829686704383621, −6.97395616544244239560997762424, −6.14800286737100168980053248466, −4.12165589787198011286358123144, −3.06100925248981481526840752106,
3.31292463029749208483972997619, 4.46926436250712479731528948633, 6.28364127429789366562515302141, 7.36359741233472302579266729911, 8.551661297107993455515236601161, 10.36119768319043165302393609982, 11.66074633018790832640286031173, 12.50415536045913476805430109353, 13.65394395825743410153663046394, 14.26663055892508374562886904958
