L(s) = 1 | + (−2.05 + 2.05i)3-s + (−2.72 − 2.72i)5-s − i·7-s − 5.44i·9-s + (0.919 + 0.919i)11-s + (1.12 − 1.12i)13-s + 11.2·15-s − 1.50·17-s + (1.46 − 1.46i)19-s + (2.05 + 2.05i)21-s + 4.77i·23-s + 9.88i·25-s + (5.02 + 5.02i)27-s + (−4.10 + 4.10i)29-s − 4.10·31-s + ⋯ |
L(s) = 1 | + (−1.18 + 1.18i)3-s + (−1.21 − 1.21i)5-s − 0.377i·7-s − 1.81i·9-s + (0.277 + 0.277i)11-s + (0.312 − 0.312i)13-s + 2.89·15-s − 0.365·17-s + (0.335 − 0.335i)19-s + (0.448 + 0.448i)21-s + 0.994i·23-s + 1.97i·25-s + (0.967 + 0.967i)27-s + (−0.761 + 0.761i)29-s − 0.738·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.273309 + 0.361735i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.273309 + 0.361735i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (2.05 - 2.05i)T - 3iT^{2} \) |
| 5 | \( 1 + (2.72 + 2.72i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.919 - 0.919i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.12 + 1.12i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + (-1.46 + 1.46i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.77iT - 23T^{2} \) |
| 29 | \( 1 + (4.10 - 4.10i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 + (-1.65 - 1.65i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.45iT - 41T^{2} \) |
| 43 | \( 1 + (-5.68 - 5.68i)T + 43iT^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 + (-0.675 - 0.675i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.13 - 1.13i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.21 + 3.21i)T - 61iT^{2} \) |
| 67 | \( 1 + (1.52 - 1.52i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 14.4iT - 73T^{2} \) |
| 79 | \( 1 - 1.77T + 79T^{2} \) |
| 83 | \( 1 + (7.16 - 7.16i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.45iT - 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−10.50155955767418270065122055511, −9.485755787718656511699799588441, −8.946414835885707570189894269063, −7.85982054957725624622403122828, −6.94612748476023534091213842602, −5.60333171074241454105627312744, −5.04284438819458301740564272322, −4.14499590285030342545938325996, −3.64965107323576048912785380606, −0.996450181514775343115952667795,
0.32799072724882432446680208482, 2.05512416542611435860937389947, 3.37562306467592144112169168015, 4.54862103925139385291771671783, 5.93516050126705452853076200558, 6.40933435484268658062995562583, 7.28569435032062072953503688039, 7.76260758572728998235950706223, 8.812755456926292808743464018755, 10.24675223656382216586752818047