| L(s) = 1 | + (0.0672 + 0.152i)2-s + (0.941 + 0.244i)3-s + (1.32 − 1.45i)4-s + (0.199 − 2.86i)5-s + (0.0260 + 0.159i)6-s + (−0.539 + 4.64i)7-s + (0.627 + 0.210i)8-s + (−1.79 − 0.998i)9-s + (0.450 − 0.162i)10-s + (5.19 − 2.44i)11-s + (1.60 − 1.04i)12-s + (0.105 − 0.0526i)13-s + (−0.744 + 0.230i)14-s + (0.889 − 2.65i)15-s + (−0.353 − 3.81i)16-s + (−2.53 + 3.25i)17-s + ⋯ |
| L(s) = 1 | + (0.0475 + 0.107i)2-s + (0.543 + 0.141i)3-s + (0.664 − 0.728i)4-s + (0.0890 − 1.28i)5-s + (0.0106 + 0.0653i)6-s + (−0.203 + 1.75i)7-s + (0.221 + 0.0743i)8-s + (−0.597 − 0.332i)9-s + (0.142 − 0.0514i)10-s + (1.56 − 0.735i)11-s + (0.464 − 0.302i)12-s + (0.0293 − 0.0146i)13-s + (−0.199 + 0.0616i)14-s + (0.229 − 0.685i)15-s + (−0.0884 − 0.954i)16-s + (−0.615 + 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70965 - 0.440448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70965 - 0.440448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (2.53 - 3.25i)T \) |
| good | 2 | \( 1 + (-0.0672 - 0.152i)T + (-1.34 + 1.47i)T^{2} \) |
| 3 | \( 1 + (-0.941 - 0.244i)T + (2.62 + 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.199 + 2.86i)T + (-4.95 - 0.690i)T^{2} \) |
| 7 | \( 1 + (0.539 - 4.64i)T + (-6.81 - 1.60i)T^{2} \) |
| 11 | \( 1 + (-5.19 + 2.44i)T + (7.02 - 8.46i)T^{2} \) |
| 13 | \( 1 + (-0.105 + 0.0526i)T + (7.83 - 10.3i)T^{2} \) |
| 19 | \( 1 + (2.63 + 1.16i)T + (12.8 + 14.0i)T^{2} \) |
| 23 | \( 1 + (-1.86 - 0.216i)T + (22.3 + 5.26i)T^{2} \) |
| 29 | \( 1 + (-2.91 - 1.05i)T + (22.3 + 18.5i)T^{2} \) |
| 31 | \( 1 + (3.10 - 3.56i)T + (-4.28 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-0.729 - 3.45i)T + (-33.8 + 14.9i)T^{2} \) |
| 41 | \( 1 + (5.69 - 3.34i)T + (19.9 - 35.8i)T^{2} \) |
| 43 | \( 1 + (-6.37 - 7.67i)T + (-7.90 + 42.2i)T^{2} \) |
| 47 | \( 1 + (-2.01 - 0.573i)T + (39.9 + 24.7i)T^{2} \) |
| 53 | \( 1 + (5.78 - 10.3i)T + (-27.9 - 45.0i)T^{2} \) |
| 59 | \( 1 + (3.90 - 0.917i)T + (52.8 - 26.2i)T^{2} \) |
| 61 | \( 1 + (2.68 - 0.438i)T + (57.8 - 19.3i)T^{2} \) |
| 67 | \( 1 + (10.4 - 4.03i)T + (49.5 - 45.1i)T^{2} \) |
| 71 | \( 1 + (-8.98 + 7.11i)T + (16.2 - 69.1i)T^{2} \) |
| 73 | \( 1 + (6.80 + 12.9i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (-0.0630 + 2.72i)T + (-78.9 - 3.64i)T^{2} \) |
| 83 | \( 1 + (-0.184 + 1.32i)T + (-79.8 - 22.7i)T^{2} \) |
| 89 | \( 1 + (-13.9 - 6.94i)T + (53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (-0.234 - 0.296i)T + (-22.2 + 94.4i)T^{2} \) |
| 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
−11.88527003245883477258572522656, −10.95242841108818441620633975738, −9.255184987376063977945188614196, −9.073209565831949003629873859917, −8.336111927232767660604924784270, −6.33368366504324801261359002201, −5.94588165511097992798076596343, −4.70789212172162417185807476962, −2.99445047577406986533953368606, −1.56650637953687619497540303351,
2.13970080454417492011396216173, 3.38808225937736985918715426351, 4.16197091518490274779622380540, 6.53228476613547591616232720144, 7.02108328071302348293025599505, 7.67762803772774376562020482105, 9.018107091400362667332344181683, 10.24822695527535436147566305421, 10.99425444410914337280681812933, 11.63503499499964550052635754289
