L(s) = 1 | + (0.276 − 0.705i)2-s + (1.34 − 1.09i)3-s + (1.04 + 0.969i)4-s + (1.69 − 0.816i)5-s + (−0.400 − 1.24i)6-s + (0.217 + 2.63i)7-s + (2.33 − 1.12i)8-s + (0.602 − 2.93i)9-s + (−0.106 − 1.42i)10-s + (−2.98 − 3.73i)11-s + (2.46 + 0.157i)12-s + (−2.16 + 5.50i)13-s + (1.92 + 0.576i)14-s + (1.38 − 2.95i)15-s + (0.0660 + 0.881i)16-s + (−1.74 + 1.61i)17-s + ⋯ |
L(s) = 1 | + (0.195 − 0.498i)2-s + (0.774 − 0.632i)3-s + (0.522 + 0.484i)4-s + (0.758 − 0.365i)5-s + (−0.163 − 0.510i)6-s + (0.0822 + 0.996i)7-s + (0.826 − 0.398i)8-s + (0.200 − 0.979i)9-s + (−0.0337 − 0.449i)10-s + (−0.898 − 1.12i)11-s + (0.711 + 0.0453i)12-s + (−0.599 + 1.52i)13-s + (0.513 + 0.154i)14-s + (0.356 − 0.762i)15-s + (0.0165 + 0.220i)16-s + (−0.422 + 0.391i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21519 - 0.978427i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21519 - 0.978427i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.34 + 1.09i)T \) |
| 7 | \( 1 + (-0.217 - 2.63i)T \) |
good | 2 | \( 1 + (-0.276 + 0.705i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.69 + 0.816i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (2.98 + 3.73i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (2.16 - 5.50i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (1.74 - 1.61i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 3.35i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.01 - 4.43i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (1.66 + 0.513i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-4.63 + 8.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.97 + 2.76i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (7.34 + 5.00i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (3.53 - 2.40i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (0.214 - 0.546i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-11.2 + 3.47i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.827 - 0.564i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-3.82 + 3.55i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (3.65 - 6.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.543 - 2.38i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-7.22 - 1.08i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.517 + 1.31i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (0.792 + 2.02i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-1.86 + 3.23i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.45228637624894169069837640356, −9.982071698808448835494971492352, −9.087554804579482556424014682721, −8.400690974835317288358959963510, −7.36293493637113136366849827485, −6.39579065297241055307674905787, −5.29094806708499601108204061163, −3.70292220477832819898308681702, −2.49222759832455225256419929278, −1.87510381518605493803912626524,
1.97861890255173175566751505990, 3.07407916997439827970444720460, 4.73403860288124103141060107092, 5.31880470173134280206450509674, 6.73815391725143453360777811812, 7.50262388819302849177556666989, 8.270856501693122649664219155588, 9.928023701593498430411523390535, 10.30087235988661321229135016683, 10.55694579248163569617160847492