L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.781 − 1.35i)3-s + (−0.499 + 0.866i)4-s + (−2.11 − 0.716i)5-s − 1.56·6-s + (−0.725 − 2.54i)7-s + 0.999·8-s + (0.278 + 0.481i)9-s + (0.438 + 2.19i)10-s + (−1.21 + 3.08i)11-s + (0.781 + 1.35i)12-s + 2.41i·13-s + (−1.84 + 1.90i)14-s + (−2.62 + 2.30i)15-s + (−0.5 − 0.866i)16-s + (−6.35 − 3.66i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.451 − 0.781i)3-s + (−0.249 + 0.433i)4-s + (−0.947 − 0.320i)5-s − 0.638·6-s + (−0.274 − 0.961i)7-s + 0.353·8-s + (0.0927 + 0.160i)9-s + (0.138 + 0.693i)10-s + (−0.366 + 0.930i)11-s + (0.225 + 0.390i)12-s + 0.670i·13-s + (−0.491 + 0.507i)14-s + (−0.677 + 0.595i)15-s + (−0.125 − 0.216i)16-s + (−1.54 − 0.889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0681013 + 0.0855544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0681013 + 0.0855544i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (2.11 + 0.716i)T \) |
| 7 | \( 1 + (0.725 + 2.54i)T \) |
| 11 | \( 1 + (1.21 - 3.08i)T \) |
good | 3 | \( 1 + (-0.781 + 1.35i)T + (-1.5 - 2.59i)T^{2} \) |
| 13 | \( 1 - 2.41iT - 13T^{2} \) |
| 17 | \( 1 + (6.35 + 3.66i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.89 + 5.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.48 - 4.32i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.47iT - 29T^{2} \) |
| 31 | \( 1 + (0.862 + 0.497i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.20 + 1.85i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 + (-5.29 - 9.17i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.54 - 0.889i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.38 + 4.26i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.71 + 2.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.76 + 5.63i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.788T + 71T^{2} \) |
| 73 | \( 1 + (8.04 + 4.64i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.68 - 1.54i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 + (3.28 - 1.89i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.37T + 97T^{2} \) |
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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
−9.589544900685405634770965270741, −8.906065283671353464258532259431, −7.85563639709298894331055261684, −7.33567838991149388463080876499, −6.71207511569296513150363112608, −4.62219509873744976389899050279, −4.23985492211982561556041394836, −2.76704865016756546941372433623, −1.67799324823671250704321318585, −0.05665691733118686825790204794,
2.50502617545140630158408365232, 3.73443016357111864812098159664, 4.42926994967464512356780328566, 5.91529289856650359834460373422, 6.39461295263251484471487254325, 7.80791363893784100500920905201, 8.522801030723648088528601112081, 8.842456182213504877782737385858, 10.17952219048830174582507086456, 10.51599909117922759088221873451