L(s) = 1 | + (1.39 + 0.258i)2-s + (1.86 + 0.719i)4-s + (−0.404 + 2.19i)5-s + 1.81·7-s + (2.40 + 1.48i)8-s + (−1.13 + 2.95i)10-s + (0.331 − 0.331i)11-s + (−0.0310 + 0.0310i)13-s + (2.52 + 0.469i)14-s + (2.96 + 2.68i)16-s + 1.00i·17-s + (−2.08 − 2.08i)19-s + (−2.33 + 3.81i)20-s + (0.547 − 0.375i)22-s − 6.22·23-s + ⋯ |
L(s) = 1 | + (0.983 + 0.183i)2-s + (0.933 + 0.359i)4-s + (−0.180 + 0.983i)5-s + 0.686·7-s + (0.851 + 0.524i)8-s + (−0.357 + 0.933i)10-s + (0.100 − 0.100i)11-s + (−0.00860 + 0.00860i)13-s + (0.674 + 0.125i)14-s + (0.740 + 0.671i)16-s + 0.243i·17-s + (−0.477 − 0.477i)19-s + (−0.522 + 0.852i)20-s + (0.116 − 0.0800i)22-s − 1.29·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.52445 + 1.45470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.52445 + 1.45470i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.404 - 2.19i)T \) |
good | 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 + (-0.331 + 0.331i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.0310 - 0.0310i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.00iT - 17T^{2} \) |
| 19 | \( 1 + (2.08 + 2.08i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 29 | \( 1 + (-6.28 - 6.28i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.11T + 31T^{2} \) |
| 37 | \( 1 + (0.0723 + 0.0723i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.06iT - 41T^{2} \) |
| 43 | \( 1 + (-3.78 - 3.78i)T + 43iT^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 + (7.04 + 7.04i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.68 + 6.68i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.89 - 2.89i)T + 61iT^{2} \) |
| 67 | \( 1 + (0.150 - 0.150i)T - 67iT^{2} \) |
| 71 | \( 1 + 14.5iT - 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + (5.48 - 5.48i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.3iT - 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−10.71176672507525405505096656900, −10.08090821523507638337446741386, −8.486290236334446011518987790084, −7.81994726312833449227059025978, −6.80079738594378858029571918575, −6.22051546998804592409020885728, −5.04589391217134455694282453755, −4.12038124804482606717221585255, −3.08211237707809972396794917866, −1.98184105447955546475241000021,
1.27102011555973442294806038860, 2.53791466990947282963088643701, 4.12518949767957865244503506960, 4.57242359879481552979979574517, 5.63619252244577107779459608934, 6.44476776315851838655948500134, 7.79371183697181418798798956449, 8.278912810493984137035371597943, 9.589678380153414920813000828120, 10.37396001162481204234921145397