| L(s) = 1 | + (0.248 + 0.968i)2-s + (−1.32 + 0.253i)3-s + (−0.876 + 0.481i)4-s + (0.448 + 2.19i)5-s + (−0.576 − 1.22i)6-s + (0.268 − 0.369i)7-s + (−0.684 − 0.728i)8-s + (−1.08 + 0.429i)9-s + (−2.01 + 0.979i)10-s + (−4.35 + 1.11i)11-s + (1.04 − 0.862i)12-s + (0.786 + 1.98i)13-s + (0.424 + 0.168i)14-s + (−1.15 − 2.79i)15-s + (0.535 − 0.844i)16-s + (2.21 − 4.03i)17-s + ⋯ |
| L(s) = 1 | + (0.175 + 0.684i)2-s + (−0.767 + 0.146i)3-s + (−0.438 + 0.240i)4-s + (0.200 + 0.979i)5-s + (−0.235 − 0.500i)6-s + (0.101 − 0.139i)7-s + (−0.242 − 0.257i)8-s + (−0.361 + 0.143i)9-s + (−0.635 + 0.309i)10-s + (−1.31 + 0.337i)11-s + (0.301 − 0.249i)12-s + (0.218 + 0.550i)13-s + (0.113 + 0.0449i)14-s + (−0.297 − 0.722i)15-s + (0.133 − 0.211i)16-s + (0.538 − 0.978i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0812846 + 0.726198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0812846 + 0.726198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.248 - 0.968i)T \) |
| 5 | \( 1 + (-0.448 - 2.19i)T \) |
| good | 3 | \( 1 + (1.32 - 0.253i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (-0.268 + 0.369i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (4.35 - 1.11i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (-0.786 - 1.98i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-2.21 + 4.03i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (1.16 - 6.12i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (8.88 + 0.559i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (-3.83 - 0.485i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (-6.79 - 3.73i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (-5.58 - 3.54i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.295 - 4.70i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-3.51 + 1.14i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-5.53 + 5.88i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-10.2 - 4.82i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (6.25 + 7.55i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (0.406 - 6.46i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (-0.269 - 2.13i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (-0.756 - 0.710i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (4.93 + 4.08i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (0.665 + 3.48i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (8.66 + 1.65i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (7.87 - 9.52i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-0.557 + 4.41i)T + (-93.9 - 24.1i)T^{2} \) |
| 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
−12.33689056832871199524880734755, −11.62146806847935663117751579532, −10.40651528147753032113220904700, −9.996816689972374025059588673755, −8.276015392880973317394922149918, −7.48449904595626758577548023598, −6.27695224426204670922434415149, −5.62062162591092975027452913303, −4.38232254010741341532612861272, −2.71921519479583536450758571412,
0.58776283529114169305999875177, 2.54152441024529887776058670532, 4.29822386859759769436237484729, 5.48330549007685118590881319604, 6.00386070432184413965089457844, 7.970023910923183496661505954500, 8.690580698571606398824218366739, 9.986753848121941052838404778485, 10.75841525387438399523474514984, 11.73447866401262459971942926465
