L(s) = 1 | + (−1.41 + 0.00754i)2-s + (−0.313 − 1.70i)3-s + (1.99 − 0.0213i)4-s + (0.752 − 2.80i)5-s + (0.455 + 2.40i)6-s + (−1.02 + 1.77i)7-s + (−2.82 + 0.0452i)8-s + (−2.80 + 1.06i)9-s + (−1.04 + 3.97i)10-s + (−0.447 − 1.67i)11-s + (−0.662 − 3.40i)12-s + (1.30 − 4.86i)13-s + (1.43 − 2.52i)14-s + (−5.02 − 0.402i)15-s + (3.99 − 0.0853i)16-s − 2.90i·17-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.00533i)2-s + (−0.180 − 0.983i)3-s + (0.999 − 0.0106i)4-s + (0.336 − 1.25i)5-s + (0.186 + 0.982i)6-s + (−0.387 + 0.671i)7-s + (−0.999 + 0.0160i)8-s + (−0.934 + 0.355i)9-s + (−0.329 + 1.25i)10-s + (−0.135 − 0.503i)11-s + (−0.191 − 0.981i)12-s + (0.361 − 1.34i)13-s + (0.384 − 0.673i)14-s + (−1.29 − 0.103i)15-s + (0.999 − 0.0213i)16-s − 0.705i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.339982 - 0.538300i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.339982 - 0.538300i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.00754i)T \) |
| 3 | \( 1 + (0.313 + 1.70i)T \) |
good | 5 | \( 1 + (-0.752 + 2.80i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (1.02 - 1.77i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.447 + 1.67i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.30 + 4.86i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + 2.90iT - 17T^{2} \) |
| 19 | \( 1 + (3.23 - 3.23i)T - 19iT^{2} \) |
| 23 | \( 1 + (0.831 - 0.480i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.48 - 5.55i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-8.00 + 4.62i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.18 + 8.18i)T - 37iT^{2} \) |
| 41 | \( 1 + (-4.06 - 7.04i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.38 - 0.907i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.60 + 2.78i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.65 - 1.65i)T + 53iT^{2} \) |
| 59 | \( 1 + (-8.27 - 2.21i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.06 + 0.284i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (9.36 + 2.50i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 2.40iT - 71T^{2} \) |
| 73 | \( 1 + 0.312iT - 73T^{2} \) |
| 79 | \( 1 + (-1.15 - 0.669i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-14.5 + 3.90i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 + (6.48 - 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.64836364723495185213952018042, −11.90956070372091313187269715250, −10.71423218425151285256770656710, −9.410928773118704156846279120470, −8.490785500893171219056380017923, −7.83759667611157254614431202749, −6.22848464915855775582003002158, −5.51265407999154563884287564791, −2.66551696015925927623311994784, −0.912648987097251171258329858713,
2.62031311066308787272523446033, 4.14674897920286938348567191603, 6.29211717740911430183253881545, 6.84827034922956150751494261240, 8.422945494547309375975244731977, 9.636888386761107643871103014948, 10.28424010572560400393537677784, 10.96925171983850410834580893286, 11.88465917754129213263855909308, 13.65764989342119935547727931962