L(s) = 1 | + (−1.19 + 1.60i)2-s + (−2.66 + 4.61i)3-s + (−1.13 − 3.83i)4-s + (1.86 − 1.07i)5-s + (−4.21 − 9.79i)6-s + (−6.91 + 1.06i)7-s + (7.50 + 2.76i)8-s + (−9.71 − 16.8i)9-s + (−0.506 + 4.28i)10-s + (−2.62 + 4.55i)11-s + (20.7 + 4.97i)12-s + 21.4i·13-s + (6.57 − 12.3i)14-s + 11.5i·15-s + (−13.4 + 8.71i)16-s + (−0.463 + 0.802i)17-s + ⋯ |
L(s) = 1 | + (−0.598 + 0.801i)2-s + (−0.888 + 1.53i)3-s + (−0.284 − 0.958i)4-s + (0.373 − 0.215i)5-s + (−0.701 − 1.63i)6-s + (−0.988 + 0.152i)7-s + (0.938 + 0.345i)8-s + (−1.07 − 1.86i)9-s + (−0.0506 + 0.428i)10-s + (−0.239 + 0.414i)11-s + (1.72 + 0.414i)12-s + 1.64i·13-s + (0.469 − 0.882i)14-s + 0.766i·15-s + (−0.838 + 0.544i)16-s + (−0.0272 + 0.0472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0605751 - 0.464002i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0605751 - 0.464002i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.19 - 1.60i)T \) |
| 7 | \( 1 + (6.91 - 1.06i)T \) |
good | 3 | \( 1 + (2.66 - 4.61i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-1.86 + 1.07i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (2.62 - 4.55i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 21.4iT - 169T^{2} \) |
| 17 | \( 1 + (0.463 - 0.802i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-2.96 - 5.13i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-7.52 + 4.34i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 9.42iT - 841T^{2} \) |
| 31 | \( 1 + (-29.8 - 17.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (11.0 - 6.40i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 43.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-39.8 + 22.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-64.5 - 37.2i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (26.8 - 46.4i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (24.0 - 13.9i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-39.2 + 67.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 74.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (16.8 - 29.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (26.1 - 15.1i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 72.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-27.4 - 47.4i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 53.7T + 9.40e3T^{2} \) |
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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
−15.85984923675904131613515803504, −15.07315018904771617394816187402, −13.73482933335553119689137149413, −11.90770023661133184854105504975, −10.56684313107544147646086255484, −9.654376591482618104117822901281, −9.039209000192181869908007681457, −6.73032678110954977339795531956, −5.62592919008448001696166003827, −4.33705001874117224469151027255,
0.60810229487898187397247251842, 2.75689121037447894498636506594, 5.78203724640360005591674670601, 7.08169332353483734518737060694, 8.219489269082987389317618063680, 10.03072664695394181000582613322, 11.02245052086254833658422747347, 12.22846238708737450351188349693, 13.01584359389768775474974410864, 13.63798351559569442129480042766