L(s) = 1 | + (−0.416 + 1.06i)2-s + (−0.936 + 1.45i)3-s + (0.513 + 0.476i)4-s + (−1.98 + 0.954i)5-s + (−1.15 − 1.59i)6-s + (−1.83 + 1.90i)7-s + (−2.77 + 1.33i)8-s + (−1.24 − 2.72i)9-s + (−0.187 − 2.49i)10-s + (1.60 + 2.00i)11-s + (−1.17 + 0.302i)12-s + (1.37 − 3.51i)13-s + (−1.25 − 2.74i)14-s + (0.464 − 3.78i)15-s + (−0.157 − 2.10i)16-s + (1.50 − 1.39i)17-s + ⋯ |
L(s) = 1 | + (−0.294 + 0.750i)2-s + (−0.540 + 0.841i)3-s + (0.256 + 0.238i)4-s + (−0.886 + 0.426i)5-s + (−0.472 − 0.653i)6-s + (−0.694 + 0.719i)7-s + (−0.980 + 0.472i)8-s + (−0.415 − 0.909i)9-s + (−0.0592 − 0.790i)10-s + (0.482 + 0.605i)11-s + (−0.339 + 0.0873i)12-s + (0.382 − 0.974i)13-s + (−0.335 − 0.732i)14-s + (0.119 − 0.976i)15-s + (−0.0393 − 0.525i)16-s + (0.363 − 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.269260 - 0.376581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.269260 - 0.376581i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.936 - 1.45i)T \) |
| 7 | \( 1 + (1.83 - 1.90i)T \) |
good | 2 | \( 1 + (0.416 - 1.06i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (1.98 - 0.954i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-1.60 - 2.00i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.37 + 3.51i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-1.50 + 1.39i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.19 + 2.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.03 - 8.89i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (4.41 + 1.36i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-1.61 + 2.79i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.05 - 1.55i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (4.74 + 3.23i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (0.447 - 0.305i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-2.06 + 5.27i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (7.50 - 2.31i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (8.58 - 5.85i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (5.50 - 5.10i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (7.35 - 12.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.89 + 8.31i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-4.24 - 0.640i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-1.58 - 2.74i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.331 + 0.844i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-3.62 - 9.23i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (6.73 - 11.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.80751295338068023811262227555, −10.96289944285178828570126988098, −9.721412632834687811817006023483, −9.135725249320804449394278082866, −7.932491701318698387362873882538, −7.18193265067277704855652144856, −6.11267283486264597465491369855, −5.40411819143747868489549481424, −3.78563400287650540178086605414, −3.02251728023006791502100989605,
0.34263840634781087731271782263, 1.57037628054918599465307707416, 3.23628385063468389699584892351, 4.40048582938438533891846493103, 6.13037895266475504109725291014, 6.57161395073349734474432659124, 7.73855732432478978497520746431, 8.694062902990743915520688795481, 9.792237324306296568115636042055, 10.83706126285251096067564688909