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 | \( 1 - 0.497T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.04030729536461974943677908461, −9.877248677073797544801568251362, −9.180444561526774944691562499763, −8.543546700711025336779351905248, −7.25928995874242580751270283277, −6.49921578498513951877438797022, −4.71489295080550718353331455968, −3.64331045904163412445564480012, −2.15272251730103728683823392625, −0.879677070960951300962396596148,
1.41905100250357820306475292076, 3.35261194610869583497469228886, 5.07293649328838668986673863175, 5.96486026604588480048041411583, 6.95860172804696422228812155509, 7.78488705159432614942585376629, 8.776973060669325797945895994141, 9.200554463623448634056392945192, 10.16567871265609252423415065497, 11.23466428902501187311298160266