L(s) = 1 | + (1.13 − 1.56i)2-s + (−0.492 − 0.159i)3-s + (−0.535 − 1.64i)4-s + (2.23 + 0.126i)5-s + (−0.809 + 0.587i)6-s + (−0.852 + 0.277i)7-s + (0.492 + 0.159i)8-s + (−2.21 − 1.60i)9-s + (2.73 − 3.34i)10-s + 0.896i·12-s + (2.49 − 3.43i)13-s + (−0.535 + 1.64i)14-s + (−1.07 − 0.419i)15-s + (3.61 − 2.62i)16-s + (−0.608 − 0.837i)17-s + (−5.01 + 1.63i)18-s + ⋯ |
L(s) = 1 | + (0.802 − 1.10i)2-s + (−0.284 − 0.0923i)3-s + (−0.267 − 0.823i)4-s + (0.998 + 0.0563i)5-s + (−0.330 + 0.239i)6-s + (−0.322 + 0.104i)7-s + (0.174 + 0.0565i)8-s + (−0.736 − 0.535i)9-s + (0.863 − 1.05i)10-s + 0.258i·12-s + (0.691 − 0.951i)13-s + (−0.143 + 0.440i)14-s + (−0.278 − 0.108i)15-s + (0.902 − 0.655i)16-s + (−0.147 − 0.203i)17-s + (−1.18 + 0.384i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41155 - 1.87277i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41155 - 1.87277i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.23 - 0.126i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.13 + 1.56i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.492 + 0.159i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (0.852 - 0.277i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-2.49 + 3.43i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.608 + 0.837i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.91 + 5.89i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 6.31iT - 23T^{2} \) |
| 29 | \( 1 + (2.14 + 6.58i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.26 - 3.09i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (6.36 - 2.06i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.535 + 1.64i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (-3.90 - 1.26i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (7.03 - 9.68i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.46 - 4.50i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (6.68 - 4.85i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 14.9iT - 67T^{2} \) |
| 71 | \( 1 + (1.77 - 1.29i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.65 - 1.51i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.42 - 3.21i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.81 - 8.00i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 + (-0.385 + 0.530i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−10.71466070201317268256118947243, −9.715057785484322654642523444124, −9.037033462552586896400285979921, −7.71631965516210829542635235299, −6.37483341580443027847901615037, −5.65516453228425152193100714058, −4.79325785306283821294205599878, −3.30075386700038427133466749998, −2.71991736222700320651164776602, −1.18795389367518958066282742735,
1.86172627932230322741698148927, 3.55669149854427866798367437551, 4.76731235148121884560682238397, 5.58104699050379626742348271986, 6.26289000760908771716544817166, 6.90528490384137864546702950086, 8.174564897714992757414315303755, 8.950112874675701116243298120141, 10.18796762346695117526335025074, 10.72563518701014801551381717333