L(s) = 1 | + (−1.16 − 2.01i)2-s + (0.436 − 1.67i)3-s + (−1.71 + 2.97i)4-s + (−3.89 + 1.07i)6-s + (2.39 − 1.11i)7-s + 3.33·8-s + (−2.61 − 1.46i)9-s + (−2.42 − 1.39i)11-s + (4.23 + 4.17i)12-s − 3.20·13-s + (−5.04 − 3.53i)14-s + (−0.459 − 0.795i)16-s + (−0.763 − 0.440i)17-s + (0.0942 + 6.99i)18-s + (−1.90 + 1.09i)19-s + ⋯ |
L(s) = 1 | + (−0.824 − 1.42i)2-s + (0.252 − 0.967i)3-s + (−0.858 + 1.48i)4-s + (−1.58 + 0.437i)6-s + (0.906 − 0.422i)7-s + 1.18·8-s + (−0.872 − 0.488i)9-s + (−0.729 − 0.421i)11-s + (1.22 + 1.20i)12-s − 0.888·13-s + (−1.34 − 0.946i)14-s + (−0.114 − 0.198i)16-s + (−0.185 − 0.106i)17-s + (0.0222 + 1.64i)18-s + (−0.436 + 0.251i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.353094 + 0.453455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.353094 + 0.453455i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.436 + 1.67i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.39 + 1.11i)T \) |
good | 2 | \( 1 + (1.16 + 2.01i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (2.42 + 1.39i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.20T + 13T^{2} \) |
| 17 | \( 1 + (0.763 + 0.440i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.90 - 1.09i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.77 + 6.53i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.15iT - 29T^{2} \) |
| 31 | \( 1 + (7.62 + 4.40i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.352 - 0.203i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.55T + 41T^{2} \) |
| 43 | \( 1 + 0.118iT - 43T^{2} \) |
| 47 | \( 1 + (-2.27 + 1.31i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.73 + 6.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 3.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.7 + 6.17i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.38 + 0.802i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.25iT - 71T^{2} \) |
| 73 | \( 1 + (0.110 - 0.192i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.56 + 2.71i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.666iT - 83T^{2} \) |
| 89 | \( 1 + (0.437 + 0.757i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6.37T + 97T^{2} \) |
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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
−10.49815778240173617968885394377, −9.385097354801705357755938436038, −8.441313093384884072451770003036, −7.942573564613574970515325459222, −6.95493185187535902924487737085, −5.45232649429870501002035874351, −3.99712993817355009436604736732, −2.64540580733690255380703542153, −1.86810088450147691338128957870, −0.40879800416285270259592670552,
2.37143474763736354307504059389, 4.26256800936823614329280583093, 5.25509745872725638553392354108, 5.82813215131111830641163222831, 7.36942270694535324334634961467, 7.87619128972870886219400575472, 8.760040572001180427634643353979, 9.481387675141112040118011539372, 10.19851337002704092950430978287, 11.14242144943371121388982732721