L(s) = 1 | + (0.602 − 0.348i)5-s + (−0.795 + 1.37i)7-s + (2.37 + 1.36i)11-s + (−4.76 + 2.75i)13-s + 5.65·17-s − 0.963i·19-s + (3.28 + 5.69i)23-s + (−2.25 + 3.91i)25-s + (−2.85 − 1.64i)29-s + (3.69 + 6.40i)31-s + 1.10i·35-s − 6.25i·37-s + (0.931 + 1.61i)41-s + (2.99 + 1.73i)43-s + (3.85 − 6.67i)47-s + ⋯ |
L(s) = 1 | + (0.269 − 0.155i)5-s + (−0.300 + 0.520i)7-s + (0.715 + 0.412i)11-s + (−1.32 + 0.763i)13-s + 1.37·17-s − 0.221i·19-s + (0.685 + 1.18i)23-s + (−0.451 + 0.782i)25-s + (−0.529 − 0.305i)29-s + (0.664 + 1.15i)31-s + 0.187i·35-s − 1.02i·37-s + (0.145 + 0.252i)41-s + (0.457 + 0.263i)43-s + (0.562 − 0.974i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28473 + 0.744091i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28473 + 0.744091i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.602 + 0.348i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.795 - 1.37i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.37 - 1.36i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.76 - 2.75i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 0.963iT - 19T^{2} \) |
| 23 | \( 1 + (-3.28 - 5.69i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.85 + 1.64i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.69 - 6.40i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.25iT - 37T^{2} \) |
| 41 | \( 1 + (-0.931 - 1.61i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.99 - 1.73i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.85 + 6.67i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2.54iT - 53T^{2} \) |
| 59 | \( 1 + (4.62 - 2.66i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.93 - 4.58i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.95 - 3.43i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 - 2.83T + 73T^{2} \) |
| 79 | \( 1 + (2.87 - 4.98i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.74 + 3.31i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 + (1.24 - 2.16i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.01288853013527228354756641497, −9.478239255154096070707989143715, −8.901556298645077981720721053553, −7.54439841489957639485469260868, −7.03807596831266291475801630241, −5.82224933413682389241995615751, −5.12501217226758796126478714317, −3.95761207863274638149643557176, −2.76609536307413729146168154094, −1.50629695029675223614074973789,
0.76982079087947000899752362364, 2.49935850593034728043975302042, 3.52931022626471289274618226604, 4.65524160644364933399926504399, 5.70992957618755319776946430769, 6.55878874442539037623169030751, 7.49998972180515225746008918309, 8.217530769364560129393573952645, 9.393468751734859057776079691867, 10.03970432066457906604721219552