L(s) = 1 | + (−1.54 − 0.889i)2-s + (−0.5 + 0.866i)3-s + (0.582 + 1.00i)4-s − 0.681i·5-s + (1.54 − 0.889i)6-s + (0.866 − 0.5i)7-s + 1.48i·8-s + (−0.499 − 0.866i)9-s + (−0.606 + 1.05i)10-s + (−0.511 − 0.295i)11-s − 1.16·12-s + (3.45 + 1.03i)13-s − 1.77·14-s + (0.590 + 0.340i)15-s + (2.48 − 4.30i)16-s + (−3.14 − 5.44i)17-s + ⋯ |
L(s) = 1 | + (−1.08 − 0.629i)2-s + (−0.288 + 0.499i)3-s + (0.291 + 0.504i)4-s − 0.304i·5-s + (0.629 − 0.363i)6-s + (0.327 − 0.188i)7-s + 0.524i·8-s + (−0.166 − 0.288i)9-s + (−0.191 + 0.332i)10-s + (−0.154 − 0.0889i)11-s − 0.336·12-s + (0.957 + 0.287i)13-s − 0.475·14-s + (0.152 + 0.0880i)15-s + (0.621 − 1.07i)16-s + (−0.762 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.505636 - 0.394869i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.505636 - 0.394869i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (-3.45 - 1.03i)T \) |
good | 2 | \( 1 + (1.54 + 0.889i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 0.681iT - 5T^{2} \) |
| 11 | \( 1 + (0.511 + 0.295i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.14 + 5.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.10 + 1.79i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.84 + 3.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.22 + 5.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.30iT - 31T^{2} \) |
| 37 | \( 1 + (9.12 + 5.26i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.136 - 0.0787i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.36 - 4.09i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + (2.44 - 1.41i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.45 + 2.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.48 - 3.74i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.05 + 3.49i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.30iT - 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 - 13.6iT - 83T^{2} \) |
| 89 | \( 1 + (10.6 + 6.12i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.18 - 5.30i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.21573148249267888158598874876, −10.89863717589719635968057935676, −9.730667692525087037931233761302, −9.046107332586196802158201865010, −8.268221994620406261328772580716, −6.96705573764341064539330542499, −5.46936274722017873550196065236, −4.42215056466928688876868233540, −2.67010648588185920453983947084, −0.841896706745642262145486861644,
1.42803683863734388698153027173, 3.53644396979686716290642718205, 5.36292458814268005653358502364, 6.53945674636936381621570568418, 7.21789236165751446897095373495, 8.418398058126957856013035298700, 8.771560322585865834559037337562, 10.28629101467921073006856250813, 10.81512218413843955499140699461, 12.06955816212330573628240899644