| L(s) = 1 | + (0.0569 − 1.41i)2-s + (−1.67 + 0.448i)3-s + (−1.99 − 0.160i)4-s + (−0.186 − 1.89i)5-s + (0.538 + 2.38i)6-s + (−0.839 − 0.561i)7-s + (−0.340 + 2.80i)8-s + (2.59 − 1.50i)9-s + (−2.68 + 0.155i)10-s + (−2.59 − 4.85i)11-s + (3.40 − 0.625i)12-s + (−0.611 + 6.21i)13-s + (−0.840 + 1.15i)14-s + (1.16 + 3.08i)15-s + (3.94 + 0.641i)16-s + (−2.40 + 5.79i)17-s + ⋯ |
| L(s) = 1 | + (0.0402 − 0.999i)2-s + (−0.965 + 0.259i)3-s + (−0.996 − 0.0804i)4-s + (−0.0834 − 0.847i)5-s + (0.219 + 0.975i)6-s + (−0.317 − 0.212i)7-s + (−0.120 + 0.992i)8-s + (0.865 − 0.500i)9-s + (−0.849 + 0.0492i)10-s + (−0.782 − 1.46i)11-s + (0.983 − 0.180i)12-s + (−0.169 + 1.72i)13-s + (−0.224 + 0.308i)14-s + (0.300 + 0.796i)15-s + (0.987 + 0.160i)16-s + (−0.582 + 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.65542\times10^{-5} + 3.01763\times10^{-5}i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65542\times10^{-5} + 3.01763\times10^{-5}i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0569 + 1.41i)T \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| good | 5 | \( 1 + (0.186 + 1.89i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (0.839 + 0.561i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (2.59 + 4.85i)T + (-6.11 + 9.14i)T^{2} \) |
| 13 | \( 1 + (0.611 - 6.21i)T + (-12.7 - 2.53i)T^{2} \) |
| 17 | \( 1 + (2.40 - 5.79i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (4.38 - 3.59i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-0.311 + 1.56i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.84 - 3.44i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (2.15 + 2.15i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.65 + 2.17i)T + (7.21 + 36.2i)T^{2} \) |
| 41 | \( 1 + (0.390 - 1.96i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.933 + 3.07i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (1.81 - 4.38i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (5.40 + 10.1i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (-0.760 - 7.72i)T + (-57.8 + 11.5i)T^{2} \) |
| 61 | \( 1 + (3.62 + 11.9i)T + (-50.7 + 33.8i)T^{2} \) |
| 67 | \( 1 + (8.02 - 2.43i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (-3.40 - 2.27i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-5.84 - 8.74i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (2.67 - 1.10i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.28 + 1.05i)T + (16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (16.0 - 3.19i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-7.95 + 7.95i)T - 97iT^{2} \) |
| 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
−10.90850562365341501708311427994, −10.02927707656436942207621523304, −8.951750775572160441388773048527, −8.325821592942996246300568428677, −6.53346550948308463617724718601, −5.58363736052588380317237821840, −4.49870510191425533718211857887, −3.72692185826655726045283158970, −1.70098721962994967690819175663, −0.00002945133898194470107310622,
2.78403148276268629877516777515, 4.62304894727922167791236433794, 5.32963753970755041009565121635, 6.45584999129139926399009756528, 7.21670166846975814759245772840, 7.77789962432795248798040296919, 9.321858986085351557019839034044, 10.24691808520131061143970016264, 10.88458179370554953769379736759, 12.23979484134345043576136889626
