| L(s) = 1 | + (−1.19 + 0.748i)2-s + (0.318 − 0.217i)3-s + (0.879 − 1.79i)4-s + (−2.23 + 0.167i)5-s + (−0.219 + 0.499i)6-s + (−2.62 − 0.318i)7-s + (0.289 + 2.81i)8-s + (−1.04 + 2.65i)9-s + (2.55 − 1.87i)10-s + (−4.62 + 1.81i)11-s + (−0.110 − 0.763i)12-s + (4.05 + 3.23i)13-s + (3.38 − 1.58i)14-s + (−0.675 + 0.538i)15-s + (−2.45 − 3.15i)16-s + (−1.83 + 1.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.848 + 0.529i)2-s + (0.184 − 0.125i)3-s + (0.439 − 0.898i)4-s + (−0.998 + 0.0748i)5-s + (−0.0897 + 0.203i)6-s + (−0.992 − 0.120i)7-s + (0.102 + 0.994i)8-s + (−0.347 + 0.884i)9-s + (0.807 − 0.591i)10-s + (−1.39 + 0.547i)11-s + (−0.0317 − 0.220i)12-s + (1.12 + 0.897i)13-s + (0.905 − 0.423i)14-s + (−0.174 + 0.139i)15-s + (−0.613 − 0.789i)16-s + (−0.445 + 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0280597 + 0.246733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0280597 + 0.246733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.19 - 0.748i)T \) |
| 7 | \( 1 + (2.62 + 0.318i)T \) |
| good | 3 | \( 1 + (-0.318 + 0.217i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (2.23 - 0.167i)T + (4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (4.62 - 1.81i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-4.05 - 3.23i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (1.83 - 1.97i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.20 + 2.08i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.13 + 2.29i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.93 + 8.48i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (0.0585 - 0.101i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.160 - 0.0494i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (4.98 - 10.3i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (1.97 + 4.09i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-10.2 - 1.54i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-2.75 - 0.850i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (0.758 - 10.1i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (2.62 + 8.51i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-5.07 - 2.92i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9.49 - 2.16i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 7.37i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (11.9 - 6.92i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.95 + 2.45i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (1.74 + 0.684i)T + (65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 1.56iT - 97T^{2} \) |
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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
−13.12769080525656103484433998229, −11.70496125383986532430676720573, −10.80807618357789610867339069991, −10.00732704776201795494493680119, −8.676685092625751106950127029643, −7.992224809273150792254701939019, −7.04802250766181684490685186030, −5.95007595541657736395751627833, −4.35845186989736974727239908425, −2.47211402794670663352728996239,
0.26228651585725270087900439901, 3.03280205363998831721849229211, 3.70093868573609931711072338019, 5.87363996315331550165698967745, 7.21072890055404698518126163787, 8.307778144503861947582871922615, 8.904705948490811522484078671641, 10.18571028739531629491172262597, 10.90983694221218018105723073487, 11.98424331517771981475810225820
