| L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−0.810 + 0.360i)5-s + (0.809 + 0.587i)6-s + (−0.287 − 2.63i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.443 + 0.768i)10-s + (−0.344 + 3.29i)11-s + (0.5 − 0.866i)12-s + (−3.78 + 2.75i)13-s + (−2.58 + 0.561i)14-s + (0.274 − 0.843i)15-s + (0.913 − 0.406i)16-s + (0.346 − 3.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (−0.386 + 0.429i)3-s + (−0.489 + 0.103i)4-s + (−0.362 + 0.161i)5-s + (0.330 + 0.239i)6-s + (−0.108 − 0.994i)7-s + (0.109 + 0.336i)8-s + (−0.0348 − 0.331i)9-s + (0.140 + 0.242i)10-s + (−0.103 + 0.994i)11-s + (0.144 − 0.249i)12-s + (−1.05 + 0.763i)13-s + (−0.691 + 0.149i)14-s + (0.0707 − 0.217i)15-s + (0.228 − 0.101i)16-s + (0.0839 − 0.798i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0791652 + 0.164298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0791652 + 0.164298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (0.287 + 2.63i)T \) |
| 11 | \( 1 + (0.344 - 3.29i)T \) |
| good | 5 | \( 1 + (0.810 - 0.360i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (3.78 - 2.75i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.346 + 3.29i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (3.78 + 0.804i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (2.93 - 5.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.05 - 6.33i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (8.97 + 3.99i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.139 - 0.154i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (1.71 + 5.27i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 + (-10.8 - 2.31i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-10.8 - 4.81i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (2.05 - 0.437i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-8.67 + 3.86i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.784 - 1.35i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.69 + 2.68i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-11.2 + 2.38i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (1.60 + 15.2i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (13.5 + 9.87i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (5.12 - 8.87i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.58 - 1.88i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−11.32001231358038284450927580042, −10.50518060312189444646143611033, −9.742152492757150182549562257904, −9.097496811337222396768955445347, −7.45232723504582671415839433307, −7.10945936904746551335719002445, −5.39390729649790388025281498062, −4.39298545242222720902751717572, −3.64229016817499905490166278096, −1.98792409520051886910533614776,
0.11452710764070423550259041797, 2.37439179679314058554569823987, 4.00716497622095979344324336014, 5.39804895204865240017604752381, 5.93772290544487389336601542835, 6.96154636414956401679802055276, 8.292107260702373543530361420571, 8.364784350759645663026330091707, 9.788726755370208703635891805036, 10.69627772486050088933053455489
