L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (−0.142 − 0.989i)6-s + (−2.28 − 0.672i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)10-s + (−2.98 + 3.44i)11-s + (0.654 − 0.755i)12-s + (−1.03 + 0.303i)13-s + (−0.991 − 2.17i)14-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.0307 + 0.213i)17-s + ⋯ |
L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.185 − 0.406i)5-s + (−0.0580 − 0.404i)6-s + (−0.865 − 0.254i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (0.303 − 0.0890i)10-s + (−0.901 + 1.04i)11-s + (0.189 − 0.218i)12-s + (−0.286 + 0.0842i)13-s + (−0.264 − 0.580i)14-s + (−0.217 + 0.139i)15-s + (−0.239 − 0.0704i)16-s + (0.00746 + 0.0518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.107881 + 0.596262i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.107881 + 0.596262i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (2.92 - 3.80i)T \) |
good | 7 | \( 1 + (2.28 + 0.672i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (2.98 - 3.44i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.03 - 0.303i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.0307 - 0.213i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.801 - 5.57i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.113 - 0.786i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (4.68 - 3.00i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.14 + 4.68i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (0.401 - 0.879i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.15 - 3.31i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 + (3.38 + 0.994i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-5.22 + 1.53i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (9.10 - 5.85i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (9.12 + 10.5i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (2.21 + 2.56i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.94 + 13.4i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (0.0557 - 0.0163i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-6.91 - 15.1i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-1.58 - 1.02i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.73 + 8.18i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.75658892170864119453902724949, −9.994864796100688018388782239034, −9.182805778669154135013828725672, −7.85995574154252911852034255548, −7.34651440233109469711881490995, −6.30503469276045190495611902890, −5.55143136941023594345931687001, −4.64078946348568784226783758547, −3.50459669223941853645908926125, −1.96420227956606014303942630178,
0.27029809948712397446854761817, 2.49147871865157318592861181780, 3.28532305348900066334745572077, 4.52430716223257622253732492741, 5.59913993006643856487797522006, 6.21395723542619278964233791049, 7.22642938940382166776108617121, 8.588083598565550153141517068163, 9.489266658850361619011099053864, 10.32024360297886287902521302241