| L(s) = 1 | + (−1.51 + 1.40i)2-s + (0.169 − 2.25i)4-s + (0.0830 − 0.211i)5-s + (1.07 + 2.41i)7-s + (0.338 + 0.424i)8-s + (0.171 + 0.436i)10-s + (4.81 + 1.48i)11-s + (−1.11 − 4.90i)13-s + (−5.02 − 2.14i)14-s + (3.36 + 0.507i)16-s + (−2.03 + 1.38i)17-s + (1.49 + 2.58i)19-s + (−0.463 − 0.223i)20-s + (−9.37 + 4.51i)22-s + (3.79 + 2.58i)23-s + ⋯ |
| L(s) = 1 | + (−1.07 + 0.993i)2-s + (0.0845 − 1.12i)4-s + (0.0371 − 0.0946i)5-s + (0.407 + 0.913i)7-s + (0.119 + 0.150i)8-s + (0.0542 + 0.138i)10-s + (1.45 + 0.447i)11-s + (−0.310 − 1.36i)13-s + (−1.34 − 0.572i)14-s + (0.841 + 0.126i)16-s + (−0.493 + 0.336i)17-s + (0.342 + 0.593i)19-s + (−0.103 − 0.0499i)20-s + (−1.99 + 0.962i)22-s + (0.791 + 0.539i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.479874 + 0.692801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.479874 + 0.692801i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.07 - 2.41i)T \) |
| good | 2 | \( 1 + (1.51 - 1.40i)T + (0.149 - 1.99i)T^{2} \) |
| 5 | \( 1 + (-0.0830 + 0.211i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-4.81 - 1.48i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (1.11 + 4.90i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (2.03 - 1.38i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.49 - 2.58i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.79 - 2.58i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (0.637 + 0.307i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (4.05 - 7.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.200 + 2.67i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-1.10 - 1.38i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (0.910 - 1.14i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (4.11 - 3.81i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.329 + 4.39i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-2.06 - 5.26i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.642 - 8.57i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-5.97 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.34 - 1.12i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (2.04 + 1.89i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-2.40 - 4.15i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.53 + 6.73i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-9.81 + 3.02i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 0.497T + 97T^{2} \) |
| 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
−11.23466428902501187311298160266, −10.16567871265609252423415065497, −9.200554463623448634056392945192, −8.776973060669325797945895994141, −7.78488705159432614942585376629, −6.95860172804696422228812155509, −5.96486026604588480048041411583, −5.07293649328838668986673863175, −3.35261194610869583497469228886, −1.41905100250357820306475292076,
0.879677070960951300962396596148, 2.15272251730103728683823392625, 3.64331045904163412445564480012, 4.71489295080550718353331455968, 6.49921578498513951877438797022, 7.25928995874242580751270283277, 8.543546700711025336779351905248, 9.180444561526774944691562499763, 9.877248677073797544801568251362, 11.04030729536461974943677908461
