L(s) = 1 | − 2.76·3-s − i·5-s + (−2.62 + 0.363i)7-s + 4.61·9-s − 3.56i·11-s + 0.710i·13-s + 2.76i·15-s + 5.25i·17-s − 7.01·19-s + (7.23 − 1.00i)21-s + 1.77i·23-s − 25-s − 4.46·27-s − 4.08·29-s + 1.18·31-s + ⋯ |
L(s) = 1 | − 1.59·3-s − 0.447i·5-s + (−0.990 + 0.137i)7-s + 1.53·9-s − 1.07i·11-s + 0.196i·13-s + 0.712i·15-s + 1.27i·17-s − 1.61·19-s + (1.57 − 0.218i)21-s + 0.370i·23-s − 0.200·25-s − 0.859·27-s − 0.757·29-s + 0.213·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4818589240\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4818589240\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (2.62 - 0.363i)T \) |
good | 3 | \( 1 + 2.76T + 3T^{2} \) |
| 11 | \( 1 + 3.56iT - 11T^{2} \) |
| 13 | \( 1 - 0.710iT - 13T^{2} \) |
| 17 | \( 1 - 5.25iT - 17T^{2} \) |
| 19 | \( 1 + 7.01T + 19T^{2} \) |
| 23 | \( 1 - 1.77iT - 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 + 7.71iT - 41T^{2} \) |
| 43 | \( 1 + 6.89iT - 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 + 0.578T + 53T^{2} \) |
| 59 | \( 1 - 6.12T + 59T^{2} \) |
| 61 | \( 1 - 8.34iT - 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 - 15.9iT - 71T^{2} \) |
| 73 | \( 1 + 2.37iT - 73T^{2} \) |
| 79 | \( 1 + 10.5iT - 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 14.5iT - 89T^{2} \) |
| 97 | \( 1 + 4.55iT - 97T^{2} \) |
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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
−8.910469251716138392044648404648, −8.520931640627388633052874406309, −7.20605549735106958796248704074, −6.44507802004840116593382606321, −5.90143665187807053751091929201, −5.39372901670164012579431350156, −4.24564753038364942566093113127, −3.54398593768865927213669524788, −1.93442548721788135424721459875, −0.54064081361871975507078928528,
0.41162471838913462884828126698, 2.04448735574771514409647271598, 3.27507439189943036730506449372, 4.45830852017247938696186160429, 5.01694095559025185111807222715, 6.06811878826347146617929730443, 6.63298381000077134312027953441, 7.04772283814065968208520219807, 8.069827267255807746268457045003, 9.426288712524689791306452113003