L(s) = 1 | + (−1.72 − 0.172i)3-s + (−1.02 + 0.594i)5-s − 1.04·7-s + (2.94 + 0.595i)9-s − 3.41i·11-s + (0.793 + 0.458i)13-s + (1.87 − 0.846i)15-s + (2.59 − 1.49i)17-s + (−4.30 + 0.688i)19-s + (1.80 + 0.180i)21-s + (1.62 + 0.940i)23-s + (−1.79 + 3.10i)25-s + (−4.96 − 1.53i)27-s + (−0.797 + 1.38i)29-s + 8.98i·31-s + ⋯ |
L(s) = 1 | + (−0.995 − 0.0997i)3-s + (−0.460 + 0.265i)5-s − 0.395·7-s + (0.980 + 0.198i)9-s − 1.02i·11-s + (0.220 + 0.127i)13-s + (0.484 − 0.218i)15-s + (0.628 − 0.363i)17-s + (−0.987 + 0.157i)19-s + (0.393 + 0.0394i)21-s + (0.339 + 0.196i)23-s + (−0.358 + 0.621i)25-s + (−0.955 − 0.295i)27-s + (−0.148 + 0.256i)29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0392 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0392 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.464849 + 0.446961i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.464849 + 0.446961i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.172i)T \) |
| 19 | \( 1 + (4.30 - 0.688i)T \) |
good | 5 | \( 1 + (1.02 - 0.594i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 + 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (-0.793 - 0.458i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.59 + 1.49i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.62 - 0.940i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.797 - 1.38i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.98iT - 31T^{2} \) |
| 37 | \( 1 - 4.54iT - 37T^{2} \) |
| 41 | \( 1 + (-0.469 - 0.813i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.73 - 4.73i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.82 - 4.51i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0418 - 0.0725i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.12 - 8.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.766 - 1.32i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.67 + 5.01i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.80 - 3.13i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.38 - 7.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.95 + 1.70i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.04iT - 83T^{2} \) |
| 89 | \( 1 + (-2.48 + 4.31i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.48 - 4.32i)T + (48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.56693033128841055051830949355, −9.581013158431873426745339860848, −8.595950331297591650079919400425, −7.61651198978984711071610610186, −6.77110709257129543149751729979, −6.02105941699806851978137666731, −5.17415559920020740235004505023, −4.02739006226397860422063723523, −3.04237992883956526382580180803, −1.19828638341361997584999276180,
0.39771299462928477408988611770, 2.08933366213181756163134216113, 3.85142627761326732502284261226, 4.48075373390127303250988230411, 5.57396729067695403394020537108, 6.37427627371658990026518549690, 7.26103667383902149932768968334, 8.076111378129726603604708860494, 9.239812998844790218618957300989, 10.01390012347207467680133757914