| L(s) = 1 | + (−1.79 − 1.12i)2-s + (−0.560 + 0.160i)3-s + (1.08 + 2.23i)4-s + (−1.94 + 1.10i)5-s + (1.18 + 0.340i)6-s + (−1.52 + 2.64i)7-s + (0.106 − 1.01i)8-s + (−2.25 + 1.40i)9-s + (4.72 + 0.205i)10-s + (0.275 − 0.305i)11-s + (−0.967 − 1.07i)12-s + (−0.127 − 3.65i)13-s + (5.71 − 3.03i)14-s + (0.913 − 0.929i)15-s + (1.72 − 2.21i)16-s + (−4.10 + 0.286i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 − 0.793i)2-s + (−0.323 + 0.0927i)3-s + (0.543 + 1.11i)4-s + (−0.870 + 0.492i)5-s + (0.484 + 0.138i)6-s + (−0.577 + 1.00i)7-s + (0.0376 − 0.358i)8-s + (−0.752 + 0.469i)9-s + (1.49 + 0.0651i)10-s + (0.0830 − 0.0922i)11-s + (−0.279 − 0.310i)12-s + (−0.0353 − 1.01i)13-s + (1.52 − 0.811i)14-s + (0.235 − 0.239i)15-s + (0.431 − 0.552i)16-s + (−0.994 + 0.0695i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.131116 - 0.198327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.131116 - 0.198327i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.94 - 1.10i)T \) |
| 19 | \( 1 + (-3.72 + 2.25i)T \) |
| good | 2 | \( 1 + (1.79 + 1.12i)T + (0.876 + 1.79i)T^{2} \) |
| 3 | \( 1 + (0.560 - 0.160i)T + (2.54 - 1.58i)T^{2} \) |
| 7 | \( 1 + (1.52 - 2.64i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.275 + 0.305i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.127 + 3.65i)T + (-12.9 + 0.906i)T^{2} \) |
| 17 | \( 1 + (4.10 - 0.286i)T + (16.8 - 2.36i)T^{2} \) |
| 23 | \( 1 + (-0.408 + 0.0574i)T + (22.1 - 6.33i)T^{2} \) |
| 29 | \( 1 + (-3.00 - 0.210i)T + (28.7 + 4.03i)T^{2} \) |
| 31 | \( 1 + (-4.28 + 1.90i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (0.480 - 1.47i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.418 + 0.535i)T + (-9.91 - 39.7i)T^{2} \) |
| 43 | \( 1 + (1.37 + 7.80i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (4.65 + 0.325i)T + (46.5 + 6.54i)T^{2} \) |
| 53 | \( 1 + (1.07 + 2.20i)T + (-32.6 + 41.7i)T^{2} \) |
| 59 | \( 1 + (1.23 + 3.06i)T + (-42.4 + 40.9i)T^{2} \) |
| 61 | \( 1 + (-0.673 + 0.0946i)T + (58.6 - 16.8i)T^{2} \) |
| 67 | \( 1 + (-3.03 - 12.1i)T + (-59.1 + 31.4i)T^{2} \) |
| 71 | \( 1 + (10.4 + 10.0i)T + (2.47 + 70.9i)T^{2} \) |
| 73 | \( 1 + (-0.0677 + 1.93i)T + (-72.8 - 5.09i)T^{2} \) |
| 79 | \( 1 + (8.37 - 2.40i)T + (66.9 - 41.8i)T^{2} \) |
| 83 | \( 1 + (-7.52 + 3.35i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.58 - 4.59i)T + (-21.5 + 86.3i)T^{2} \) |
| 97 | \( 1 + (-2.80 + 11.2i)T + (-85.6 - 45.5i)T^{2} \) |
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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
−10.69386589541806306369341087537, −9.994875458733404158093801183385, −8.866967369776339449321849637777, −8.380762414922813967556846774817, −7.40806163678963154221194207512, −6.15959775043756486960303233341, −5.01505497366533087033951605366, −3.18198821090946188792406925625, −2.51343281873886084890398906656, −0.29374360677077099756655058097,
0.996026444130158387214186447909, 3.49610356628663193276952738348, 4.63796474395333872844795582808, 6.21845869732647499004033180269, 6.87547201459608131657846146947, 7.66239887456072791892625902302, 8.608527028881935877691091272081, 9.264433117860930075723680865580, 10.13740761921110198549990158712, 11.18457581513920723598177234149
