L(s) = 1 | + (−0.701 + 1.78i)2-s + (0.322 − 1.70i)3-s + (−1.23 − 1.14i)4-s + (0.740 − 0.356i)5-s + (2.81 + 1.76i)6-s + (0.756 − 2.53i)7-s + (−0.547 + 0.263i)8-s + (−2.79 − 1.09i)9-s + (0.117 + 1.57i)10-s + (1.56 + 1.96i)11-s + (−2.34 + 1.73i)12-s + (−0.227 + 0.579i)13-s + (3.99 + 3.12i)14-s + (−0.368 − 1.37i)15-s + (−0.338 − 4.52i)16-s + (2.66 − 2.47i)17-s + ⋯ |
L(s) = 1 | + (−0.495 + 1.26i)2-s + (0.186 − 0.982i)3-s + (−0.617 − 0.572i)4-s + (0.331 − 0.159i)5-s + (1.14 + 0.722i)6-s + (0.285 − 0.958i)7-s + (−0.193 + 0.0931i)8-s + (−0.930 − 0.366i)9-s + (0.0372 + 0.497i)10-s + (0.473 + 0.593i)11-s + (−0.677 + 0.499i)12-s + (−0.0630 + 0.160i)13-s + (1.06 + 0.836i)14-s + (−0.0950 − 0.355i)15-s + (−0.0846 − 1.13i)16-s + (0.646 − 0.600i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16449 - 0.103932i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16449 - 0.103932i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.322 + 1.70i)T \) |
| 7 | \( 1 + (-0.756 + 2.53i)T \) |
good | 2 | \( 1 + (0.701 - 1.78i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-0.740 + 0.356i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-1.56 - 1.96i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.227 - 0.579i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-2.66 + 2.47i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.83 + 6.64i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.990 + 4.34i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (1.23 + 0.382i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-5.56 + 9.63i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.48 + 0.458i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-8.03 - 5.48i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (0.484 - 0.330i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (4.22 - 10.7i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-7.24 + 2.23i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (3.85 - 2.62i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (5.30 - 4.92i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (5.46 - 9.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.970 + 4.25i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.18 - 0.328i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-2.70 - 4.67i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.62 + 6.67i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (0.290 + 0.739i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (3.44 - 5.96i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.25966009239152781623461272704, −9.677648502898649130197259676083, −9.152817208875647207239264921226, −7.945173333742679297025420603533, −7.40890269987247251608632495572, −6.75252208865965542089509868141, −5.81290056160602861737244566596, −4.60461924967738891692449482194, −2.73421735264793237982764909812, −0.919944641701424301050506025535,
1.68338467562373143735302053333, 3.01879798476375013991547145545, 3.75432660227490131074150182615, 5.37523416430509323095572561875, 6.09809703754945038651710849792, 8.069496098996945170595973980904, 8.790127251502890525497916821313, 9.618169701764308979663639432877, 10.22067677723505321008590467042, 10.96932238758896227631228253113