| L(s) = 1 | + (−0.912 − 0.622i)2-s + (−0.609 + 1.62i)3-s + (−0.284 − 0.725i)4-s + (−1.69 − 1.57i)5-s + (1.56 − 1.10i)6-s + (−1.84 − 1.90i)7-s + (−0.683 + 2.99i)8-s + (−2.25 − 1.97i)9-s + (0.568 + 2.48i)10-s + (2.79 + 1.90i)11-s + (1.35 − 0.0198i)12-s + (−0.0571 + 0.763i)13-s + (0.496 + 2.88i)14-s + (3.58 − 1.78i)15-s + (1.34 − 1.24i)16-s + (4.56 + 5.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 0.440i)2-s + (−0.351 + 0.936i)3-s + (−0.142 − 0.362i)4-s + (−0.757 − 0.703i)5-s + (0.638 − 0.449i)6-s + (−0.695 − 0.718i)7-s + (−0.241 + 1.05i)8-s + (−0.752 − 0.658i)9-s + (0.179 + 0.787i)10-s + (0.842 + 0.574i)11-s + (0.389 − 0.00573i)12-s + (−0.0158 + 0.211i)13-s + (0.132 + 0.769i)14-s + (0.924 − 0.462i)15-s + (0.335 − 0.311i)16-s + (1.10 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.357437 + 0.232760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.357437 + 0.232760i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.609 - 1.62i)T \) |
| 7 | \( 1 + (1.84 + 1.90i)T \) |
| good | 2 | \( 1 + (0.912 + 0.622i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (1.69 + 1.57i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (-2.79 - 1.90i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (0.0571 - 0.763i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-4.56 - 5.71i)T + (-3.78 + 16.5i)T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 + (-0.0694 - 0.176i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (2.59 - 6.60i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (4.84 - 8.39i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.37 - 1.72i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (1.77 + 1.64i)T + (3.06 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-5.35 + 4.96i)T + (3.21 - 42.8i)T^{2} \) |
| 47 | \( 1 + (-7.35 - 5.01i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (5.33 - 6.69i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + (6.38 + 1.97i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (1.91 - 4.88i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-5.88 + 10.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.46 + 5.60i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-3.02 - 1.45i)T + (45.5 + 57.0i)T^{2} \) |
| 79 | \( 1 + (1.77 + 3.07i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.847 - 11.3i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-6.74 - 3.24i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-7.80 - 13.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.72600230512736943434100902716, −10.61603223227252246306881369715, −9.426403779951218137899941112387, −9.019617726021358779695924680447, −7.940464258091783675127341381693, −6.55092039257969641522858521050, −5.41510144173969242697427278665, −4.33178290787320500114745716533, −3.53297588375360149613144585598, −1.23072074853396361434315776642,
0.40407679862101513415314987325, 2.76528333153072783337893059083, 3.76959077562246472108371144968, 5.69201467780346354397617517614, 6.53248362258973441844721924109, 7.40592696950047818999685260958, 7.932878318241853354449063905302, 9.022436787628868725820594938093, 9.774468689969915582602153243246, 11.29541251559248039266419153519
