| L(s) = 1 | + i·2-s + i·3-s − 4-s + (1.88 + 1.19i)5-s − 6-s − i·8-s − 9-s + (−1.19 + 1.88i)10-s + 0.979·11-s − i·12-s + 0.435i·13-s + (−1.19 + 1.88i)15-s + 16-s + 2.79i·17-s − i·18-s + 7.34·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.844 + 0.535i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (−0.378 + 0.597i)10-s + 0.295·11-s − 0.288i·12-s + 0.120i·13-s + (−0.308 + 0.487i)15-s + 0.250·16-s + 0.678i·17-s − 0.235i·18-s + 1.68·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.768139429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.768139429\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.88 - 1.19i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 0.979T + 11T^{2} \) |
| 13 | \( 1 - 0.435iT - 13T^{2} \) |
| 17 | \( 1 - 2.79iT - 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 23 | \( 1 - 3.34iT - 23T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 + 4.97T + 31T^{2} \) |
| 37 | \( 1 - 4.63iT - 37T^{2} \) |
| 41 | \( 1 + 4.94T + 41T^{2} \) |
| 43 | \( 1 - 9.97iT - 43T^{2} \) |
| 47 | \( 1 + 4.40iT - 47T^{2} \) |
| 53 | \( 1 + 0.657iT - 53T^{2} \) |
| 59 | \( 1 - 8.27T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 2.36iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 15.3iT - 73T^{2} \) |
| 79 | \( 1 - 2.88T + 79T^{2} \) |
| 83 | \( 1 - 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−9.638289492538858228496729979267, −9.244362712334040226871554780422, −8.211427468331300645316899484071, −7.31161370861793620958370446259, −6.55628928671433275296387150080, −5.64346712122864626868399714542, −5.15890741101407074324519463473, −3.88315030898896143195924659899, −3.06717957679776381207045672986, −1.55981276824074968851753392760,
0.74520111361372945796411256194, 1.80792118094520052041080989750, 2.77442544811803709647213590551, 3.89978592285212749655182824165, 5.19348476079307132490218072865, 5.59602748266858758874661632719, 6.79394981004993145251898903757, 7.58025771254779245744554120975, 8.640697312426262205478850507915, 9.239014576076508604822602104406
