L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (1.62 + 2.09i)7-s + 0.999i·8-s + (−2.59 − 1.5i)11-s + 2.44i·13-s + (−2.44 − 0.999i)14-s + (−0.5 − 0.866i)16-s + (0.507 − 0.878i)17-s + (−0.878 + 0.507i)19-s + 3·22-s + (3.67 − 2.12i)23-s + (−1.22 − 2.12i)26-s + (2.62 − 0.358i)28-s − 1.24i·29-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.612 + 0.790i)7-s + 0.353i·8-s + (−0.783 − 0.452i)11-s + 0.679i·13-s + (−0.654 − 0.267i)14-s + (−0.125 − 0.216i)16-s + (0.123 − 0.213i)17-s + (−0.201 + 0.116i)19-s + 0.639·22-s + (0.766 − 0.442i)23-s + (−0.240 − 0.416i)26-s + (0.495 − 0.0677i)28-s − 0.230i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.105779096\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.105779096\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.62 - 2.09i)T \) |
good | 11 | \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (-0.507 + 0.878i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.878 - 0.507i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.24iT - 29T^{2} \) |
| 31 | \( 1 + (-4.86 - 2.80i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.12 - 7.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + (-0.507 - 0.878i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.07 + 0.621i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.76 - 9.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.12 + 2.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 + (7.24 + 4.18i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.62 - 9.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.76iT - 97T^{2} \) |
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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
−8.752742031851980234056398951953, −8.263116819859376050327486846572, −7.59642918319293815720609980552, −6.65345266803831546153183471778, −6.01626644500107315000586191836, −5.12037259095818188952337002888, −4.55067323334301733754103326046, −3.08422113787074866686839713766, −2.29022266079799523413965184335, −1.14657511021093436113004249526,
0.46521301317213635668169933435, 1.60402582523411515242624414103, 2.64053639490025127341516391349, 3.57809082300480419805992260702, 4.55496263285553851619221873351, 5.27169503413888698179441377210, 6.32554835365969881968027761586, 7.29751596520034587946887054675, 7.74017839819997984179307168047, 8.366516899476491193873984960053