| L(s) = 1 | + (−2.21 − 0.870i)2-s + (2.69 + 2.50i)4-s + (−1.55 − 1.06i)5-s + (0.00333 − 2.64i)7-s + (−1.73 − 3.60i)8-s + (2.53 + 3.71i)10-s + (−0.0972 − 0.645i)11-s + (0.531 − 0.424i)13-s + (−2.31 + 5.86i)14-s + (0.163 + 2.18i)16-s + (5.58 + 1.72i)17-s + (−5.58 + 3.22i)19-s + (−1.54 − 6.76i)20-s + (−0.346 + 1.51i)22-s + (−2.20 − 7.15i)23-s + ⋯ |
| L(s) = 1 | + (−1.56 − 0.615i)2-s + (1.34 + 1.25i)4-s + (−0.696 − 0.474i)5-s + (0.00125 − 0.999i)7-s + (−0.614 − 1.27i)8-s + (0.800 + 1.17i)10-s + (−0.0293 − 0.194i)11-s + (0.147 − 0.117i)13-s + (−0.617 + 1.56i)14-s + (0.0408 + 0.545i)16-s + (1.35 + 0.417i)17-s + (−1.28 + 0.739i)19-s + (−0.345 − 1.51i)20-s + (−0.0737 + 0.323i)22-s + (−0.460 − 1.49i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0393680 + 0.229223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0393680 + 0.229223i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.00333 + 2.64i)T \) |
| good | 2 | \( 1 + (2.21 + 0.870i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (1.55 + 1.06i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.0972 + 0.645i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-0.531 + 0.424i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-5.58 - 1.72i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (5.58 - 3.22i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.20 + 7.15i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (4.23 - 0.966i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.43 + 1.40i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.42 - 2.25i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (3.48 - 1.67i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (2.03 + 0.982i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (1.26 - 3.22i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (6.96 - 7.50i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-1.50 + 1.02i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (6.85 + 7.39i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (5.80 - 10.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (15.2 + 3.47i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.21 + 3.22i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-4.60 - 7.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.98 + 7.50i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.81 - 0.725i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 8.08iT - 97T^{2} \) |
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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
−10.46021406013711723516756325644, −9.928366589311349932933296852625, −8.690748891255380348148120984440, −8.103292162740700175953315503818, −7.47352471814080676437032955073, −6.21973749474192974380640405549, −4.43267845813089749053301706907, −3.34869013907078817966445408809, −1.62544679798569438119574055663, −0.24445632946568512817961565947,
1.87781125092402283749468616935, 3.48534965257203583753120551554, 5.33427664616841940121945120762, 6.34908720417004345887653257789, 7.36512195312227189019711519643, 7.934473231815374341500397863579, 8.925695104945037249154491403161, 9.540953133669145133636982822576, 10.52171538262529115511373494981, 11.38397205753083827284904126651
