| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−1.08 + 2.38i)3-s + (−0.959 + 0.281i)4-s + (0.443 − 0.512i)5-s + (2.51 + 0.739i)6-s + (−0.841 + 0.540i)7-s + (0.415 + 0.909i)8-s + (−2.54 − 2.93i)9-s + (−0.569 − 0.366i)10-s + (−0.529 + 3.68i)11-s + (0.373 − 2.59i)12-s + (0.0806 + 0.0518i)13-s + (0.654 + 0.755i)14-s + (0.738 + 1.61i)15-s + (0.841 − 0.540i)16-s + (−7.52 − 2.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.629 + 1.37i)3-s + (−0.479 + 0.140i)4-s + (0.198 − 0.228i)5-s + (1.02 + 0.301i)6-s + (−0.317 + 0.204i)7-s + (0.146 + 0.321i)8-s + (−0.847 − 0.977i)9-s + (−0.180 − 0.115i)10-s + (−0.159 + 1.11i)11-s + (0.107 − 0.749i)12-s + (0.0223 + 0.0143i)13-s + (0.175 + 0.201i)14-s + (0.190 + 0.417i)15-s + (0.210 − 0.135i)16-s + (−1.82 − 0.535i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.228722 + 0.488093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228722 + 0.488093i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (-4.07 - 2.52i)T \) |
| good | 3 | \( 1 + (1.08 - 2.38i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.443 + 0.512i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.529 - 3.68i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.0806 - 0.0518i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (7.52 + 2.20i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (7.32 - 2.15i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (3.73 + 1.09i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.242 + 0.530i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.26 - 4.92i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.00 + 3.47i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.547 + 1.19i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + (-4.71 + 3.03i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (0.235 + 0.151i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.64 - 5.79i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.21 - 15.4i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.800 + 5.56i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (4.11 - 1.20i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (12.7 + 8.20i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-10.1 - 11.7i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.279 + 0.611i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (4.10 - 4.74i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.63775264788850873214978232208, −10.92450669157320912635741139440, −10.21264227832306153134164637377, −9.349258258310398531293948257479, −8.810344427451996648772541512622, −7.07287337141798412445088760149, −5.70673249444634605184108307766, −4.66053801709701546677402144874, −4.00575522265361037204397062138, −2.31752316498821086293315156977,
0.40959955859834572998435046716, 2.34808446162304431745323474446, 4.35361678165936203274489874846, 5.91643142653035493585474921630, 6.43549494686813108021038822844, 7.13187175027520800983312190711, 8.316570434329833695948353517096, 9.018202425301303128573663587655, 10.74595267900737405512375984271, 11.09694462917123346035268058014
