L(s) = 1 | + (−1.79 + 2.18i)2-s + (−1.53 − 7.85i)4-s + 17.7i·5-s + (5.15 − 17.7i)7-s + (19.8 + 10.7i)8-s + (−38.7 − 31.9i)10-s + 10.7i·11-s + 72.4i·13-s + (29.5 + 43.2i)14-s + (−59.3 + 24.0i)16-s − 62.7i·17-s + 98.1·19-s + (139. − 27.2i)20-s + (−23.5 − 19.3i)22-s + 160. i·23-s + ⋯ |
L(s) = 1 | + (−0.635 + 0.771i)2-s + (−0.191 − 0.981i)4-s + 1.58i·5-s + (0.278 − 0.960i)7-s + (0.879 + 0.476i)8-s + (−1.22 − 1.01i)10-s + 0.295i·11-s + 1.54i·13-s + (0.564 + 0.825i)14-s + (−0.926 + 0.375i)16-s − 0.895i·17-s + 1.18·19-s + (1.56 − 0.304i)20-s + (−0.228 − 0.187i)22-s + 1.45i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0891i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8806617442\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8806617442\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.79 - 2.18i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-5.15 + 17.7i)T \) |
good | 5 | \( 1 - 17.7iT - 125T^{2} \) |
| 11 | \( 1 - 10.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 72.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 62.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 98.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 160. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 90.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 139.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 297. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 22.8iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 484.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 502.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 148.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 438. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 667. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 174. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 680. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 780.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 27.2iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 212. iT - 9.12e5T^{2} \) |
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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
−11.34394747671598558421023006701, −11.23745729465486699817524406541, −9.879660632127541991949622458286, −9.435150082041889402499747975153, −7.70797107932457678078757202139, −7.17669274577168675966437942233, −6.50441467837979172565507686650, −5.03039703731686838791927784666, −3.55110748215450488026193195023, −1.70578543789584396373929584182,
0.43915067889361789248598756622, 1.75532572344079101547248808512, 3.32726754199789277399284275224, 4.83005411554370129251576048528, 5.72313316919103475373178424490, 7.73452888838170503753301730966, 8.503641240014384411248245683704, 9.037593663145907851876492544024, 10.11532746428358138456006290377, 11.15701882962208254833156256073