| L(s) = 1 | + (1.64 + 0.541i)3-s + (0.523 − 1.43i)5-s + (0.0694 + 0.0122i)7-s + (2.41 + 1.78i)9-s + (−0.697 + 0.253i)11-s + (0.808 − 0.678i)13-s + (1.63 − 2.08i)15-s + (3.11 − 1.79i)17-s + (2.27 + 1.31i)19-s + (0.107 + 0.0576i)21-s + (−0.0893 − 0.506i)23-s + (2.03 + 1.71i)25-s + (3.00 + 4.23i)27-s + (2.39 − 2.84i)29-s + (1.20 − 0.212i)31-s + ⋯ |
| L(s) = 1 | + (0.949 + 0.312i)3-s + (0.233 − 0.642i)5-s + (0.0262 + 0.00462i)7-s + (0.804 + 0.593i)9-s + (−0.210 + 0.0765i)11-s + (0.224 − 0.188i)13-s + (0.422 − 0.537i)15-s + (0.754 − 0.435i)17-s + (0.522 + 0.301i)19-s + (0.0234 + 0.0125i)21-s + (−0.0186 − 0.105i)23-s + (0.407 + 0.342i)25-s + (0.578 + 0.815i)27-s + (0.443 − 0.529i)29-s + (0.216 − 0.0381i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.36505 - 0.0610001i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.36505 - 0.0610001i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.64 - 0.541i)T \) |
| good | 5 | \( 1 + (-0.523 + 1.43i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.0694 - 0.0122i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.697 - 0.253i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.808 + 0.678i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.11 + 1.79i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.27 - 1.31i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0893 + 0.506i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.39 + 2.84i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.20 + 0.212i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (3.22 + 5.58i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.806 + 0.961i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.81 - 7.73i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.451 - 2.56i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 12.1iT - 53T^{2} \) |
| 59 | \( 1 + (7.77 + 2.83i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.200 + 1.13i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.67 - 4.38i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.52 - 7.83i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.79 - 6.57i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.02 - 10.7i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (9.38 + 7.87i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-8.71 - 5.02i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.929 + 0.338i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.859689341914140793860785692080, −9.416637985708395560725804669217, −8.435587223614158532495251924219, −7.87823207630620997841905023265, −6.89415710736760047922994700772, −5.55870289891567752136908090386, −4.78827926235176631469053210468, −3.69023363809313788301832250987, −2.68739109325567543904623020648, −1.32740809114548612428539549972,
1.43496009400380489289697413846, 2.74699042019715057366145662250, 3.46095418974566945440163974532, 4.71249406719242802411126072791, 6.01685148942845789096015013162, 6.86811766613434799030409722729, 7.63826626461893341617928427740, 8.472512165044998087256457294194, 9.262694333689233960174788305800, 10.16313976777630837779086676740
