| L(s) = 1 | + (−1.36 + 0.370i)2-s + (0.540 − 0.841i)3-s + (1.72 − 1.01i)4-s + (0.808 + 0.369i)5-s + (−0.426 + 1.34i)6-s + (0.759 + 0.223i)7-s + (−1.98 + 2.01i)8-s + (−0.415 − 0.909i)9-s + (−1.24 − 0.204i)10-s + (0.857 + 0.742i)11-s + (0.0824 − 1.99i)12-s + (1.80 + 6.16i)13-s + (−1.11 − 0.0230i)14-s + (0.748 − 0.480i)15-s + (1.95 − 3.48i)16-s + (0.262 + 1.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.965 + 0.261i)2-s + (0.312 − 0.485i)3-s + (0.862 − 0.505i)4-s + (0.361 + 0.165i)5-s + (−0.174 + 0.550i)6-s + (0.287 + 0.0843i)7-s + (−0.700 + 0.713i)8-s + (−0.138 − 0.303i)9-s + (−0.392 − 0.0647i)10-s + (0.258 + 0.223i)11-s + (0.0238 − 0.576i)12-s + (0.501 + 1.70i)13-s + (−0.299 − 0.00617i)14-s + (0.193 − 0.124i)15-s + (0.488 − 0.872i)16-s + (0.0637 + 0.443i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17838 + 0.211607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17838 + 0.211607i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 - 0.370i)T \) |
| 3 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (4.76 - 0.573i)T \) |
| good | 5 | \( 1 + (-0.808 - 0.369i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.759 - 0.223i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.857 - 0.742i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.80 - 6.16i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.262 - 1.82i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-6.71 - 0.965i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.980 - 0.140i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.39 + 2.18i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-6.20 + 2.83i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.147 + 0.323i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.14 + 3.33i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + (2.70 - 9.21i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (2.36 + 8.03i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.32 - 6.73i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (9.22 - 7.99i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-1.86 - 2.15i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.472 + 3.28i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-6.92 + 2.03i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-11.7 + 5.35i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-1.13 - 0.726i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (3.20 - 7.02i)T + (-63.5 - 73.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.70462472200343340966764520260, −9.636869174680799492299521556329, −9.174196504533074377658084252498, −8.132512445600670977132118461432, −7.39595599387200054073995574924, −6.45029021259420937724851262013, −5.73703647130334022918234556328, −4.05468181400157783290689109742, −2.39063084349100053035190035403, −1.41874904727271603058753014673,
1.07691540738187535447706025730, 2.71602334428493558235159157724, 3.64793682315531913900838050565, 5.24699417974531871374206603460, 6.18273442371816498879275792537, 7.65299056916871225917052563299, 8.055929661323188863412729820479, 9.176154112490963941351551780872, 9.768656445479133243086956432884, 10.55270444569758658007852099896
