L(s) = 1 | + (−0.908 − 1.08i)2-s + (−0.348 + 1.96i)4-s + (−2.24 − 1.29i)5-s + (−0.866 + 0.5i)7-s + (2.45 − 1.41i)8-s + (0.635 + 3.60i)10-s + (0.124 + 0.216i)11-s + (−0.0646 + 0.111i)13-s + (1.32 + 0.484i)14-s + (−3.75 − 1.37i)16-s − 0.554i·17-s + 3.58i·19-s + (3.33 − 3.96i)20-s + (0.120 − 0.331i)22-s + (−3.94 + 6.83i)23-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.174 + 0.984i)4-s + (−1.00 − 0.579i)5-s + (−0.327 + 0.188i)7-s + (0.866 − 0.499i)8-s + (0.200 + 1.14i)10-s + (0.0376 + 0.0651i)11-s + (−0.0179 + 0.0310i)13-s + (0.355 + 0.129i)14-s + (−0.939 − 0.342i)16-s − 0.134i·17-s + 0.823i·19-s + (0.744 − 0.886i)20-s + (0.0257 − 0.0707i)22-s + (−0.823 + 1.42i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.682309 + 0.0576456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.682309 + 0.0576456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.908 + 1.08i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 + (2.24 + 1.29i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.124 - 0.216i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0646 - 0.111i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 0.554iT - 17T^{2} \) |
| 19 | \( 1 - 3.58iT - 19T^{2} \) |
| 23 | \( 1 + (3.94 - 6.83i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.90 + 3.40i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.96 - 4.02i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.96T + 37T^{2} \) |
| 41 | \( 1 + (4.25 + 2.45i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.78 + 2.18i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.41 - 4.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 9.00iT - 53T^{2} \) |
| 59 | \( 1 + (3.71 - 6.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.42 - 11.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.23 - 3.02i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + (-2.04 + 1.17i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.36 - 4.09i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 7.82iT - 89T^{2} \) |
| 97 | \( 1 + (5.67 + 9.82i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.23547216849711308989284601561, −9.624403855769584073945876619610, −8.629491716934420126097259593969, −8.021781489030598832957405980464, −7.28886878738043883783618367039, −5.96975194097338020180855911637, −4.52562572112666222344376762261, −3.82500804469334230967727917293, −2.66168234948920160391965753726, −1.07170220648840418017056846471,
0.54363990453613199050546736402, 2.59284378311707346642825685806, 4.01284786875429801784550929967, 4.94213856597505099802203471121, 6.40508491936550558733566626772, 6.73859871020144156564682373107, 7.908637010409150145530152432785, 8.282296693344070313136357676164, 9.439675147022665405990163107299, 10.22279318227004303191430284722