L(s) = 1 | + (1.97 + 1.14i)5-s + (0.907 + 1.57i)7-s + (−4.24 + 2.44i)11-s + (4.00 + 2.31i)13-s − 1.92·17-s + 2.12i·19-s + (−1.15 + 2.00i)23-s + (0.101 + 0.175i)25-s + (−3.16 + 1.82i)29-s + (2.65 − 4.60i)31-s + 4.14i·35-s + 7.98i·37-s + (2.36 − 4.09i)41-s + (2.20 − 1.27i)43-s + (−2.02 − 3.49i)47-s + ⋯ |
L(s) = 1 | + (0.883 + 0.510i)5-s + (0.343 + 0.594i)7-s + (−1.27 + 0.738i)11-s + (1.11 + 0.641i)13-s − 0.467·17-s + 0.488i·19-s + (−0.241 + 0.418i)23-s + (0.0203 + 0.0351i)25-s + (−0.587 + 0.339i)29-s + (0.477 − 0.826i)31-s + 0.700i·35-s + 1.31i·37-s + (0.368 − 0.639i)41-s + (0.336 − 0.194i)43-s + (−0.294 − 0.510i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32212 + 1.05468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32212 + 1.05468i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.97 - 1.14i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.907 - 1.57i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.24 - 2.44i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.00 - 2.31i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 - 2.12iT - 19T^{2} \) |
| 23 | \( 1 + (1.15 - 2.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.16 - 1.82i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.65 + 4.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.98iT - 37T^{2} \) |
| 41 | \( 1 + (-2.36 + 4.09i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 + 1.27i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.02 + 3.49i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.95iT - 53T^{2} \) |
| 59 | \( 1 + (-3.05 - 1.76i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.71 + 0.991i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.72 - 4.46i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + (-4.97 - 8.61i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.12 + 1.80i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 2.49T + 89T^{2} \) |
| 97 | \( 1 + (-6.99 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.28688340384797166823084244472, −9.625253827500442987671806461537, −8.673916461657845651389015456446, −7.86312789850958160922168219774, −6.80996182849673017901622750793, −5.94692799005992959587260811851, −5.23215631221696676393845904312, −4.02737777251178319860462922282, −2.59958213555853655567872268630, −1.82140460278598427057554016823,
0.828042184616979391816122185887, 2.25790686195285850062392514452, 3.51276104369490012064004039128, 4.78173998459225439272983055815, 5.58885772985189031163594860349, 6.32975532806085916825065001929, 7.59375475021026998109055011459, 8.329196053360236695888143430698, 9.082565850190104227955838635630, 10.08015607923296985531685838058