| L(s) = 1 | + (0.934 − 2.38i)2-s + (1.38 − 1.04i)3-s + (−3.33 − 3.08i)4-s + (−2.46 + 1.18i)5-s + (−1.18 − 4.26i)6-s + (−2.60 + 0.461i)7-s + (−5.85 + 2.82i)8-s + (0.830 − 2.88i)9-s + (0.523 + 6.98i)10-s + (−0.789 − 0.990i)11-s + (−7.82 − 0.808i)12-s + (0.630 − 1.60i)13-s + (−1.33 + 6.63i)14-s + (−2.17 + 4.21i)15-s + (0.564 + 7.52i)16-s + (3.06 − 2.84i)17-s + ⋯ |
| L(s) = 1 | + (0.660 − 1.68i)2-s + (0.799 − 0.601i)3-s + (−1.66 − 1.54i)4-s + (−1.10 + 0.531i)5-s + (−0.484 − 1.74i)6-s + (−0.984 + 0.174i)7-s + (−2.07 + 0.997i)8-s + (0.276 − 0.960i)9-s + (0.165 + 2.20i)10-s + (−0.238 − 0.298i)11-s + (−2.25 − 0.233i)12-s + (0.174 − 0.445i)13-s + (−0.357 + 1.77i)14-s + (−0.562 + 1.08i)15-s + (0.141 + 1.88i)16-s + (0.743 − 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.748 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.519128 + 1.37015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.519128 + 1.37015i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.38 + 1.04i)T \) |
| 7 | \( 1 + (2.60 - 0.461i)T \) |
| good | 2 | \( 1 + (-0.934 + 2.38i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (2.46 - 1.18i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (0.789 + 0.990i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.630 + 1.60i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-3.06 + 2.84i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.28 + 3.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.545 - 2.38i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-3.60 - 1.11i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-2.18 + 3.78i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (11.3 + 3.50i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-8.81 - 6.01i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-7.19 + 4.90i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-0.446 + 1.13i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (6.86 - 2.11i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (5.14 - 3.50i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-9.90 + 9.18i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (2.03 - 3.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.44 - 10.7i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-15.7 - 2.36i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-1.41 - 2.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.15 - 10.5i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (5.94 + 15.1i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (4.17 - 7.22i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.82346085240918897408798536804, −9.794581824598056544628550268277, −9.116878495635409972717469158391, −7.905862635717988101325472675865, −6.91700013031655593996471542712, −5.50334842699257157590177669262, −3.97497163432114009529633537782, −3.22064409171798952647871761138, −2.64200385589635784772089031858, −0.70178964661941026462831079364,
3.36329974278006019082344884970, 4.02696836681065406601112498756, 4.88369611217269249680849445688, 6.06750043502518521812817866941, 7.19563193317629038530153122126, 7.944605216476893957561880690769, 8.548309286768846233761576513751, 9.475716200673222253223219331670, 10.51340665021725452667044253906, 12.31140240124232521965345340195
