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 | \( 1 + (7.48 + 4.32i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.01390012347207467680133757914, −9.239812998844790218618957300989, −8.076111378129726603604708860494, −7.26103667383902149932768968334, −6.37427627371658990026518549690, −5.57396729067695403394020537108, −4.48075373390127303250988230411, −3.85142627761326732502284261226, −2.08933366213181756163134216113, −0.39771299462928477408988611770,
1.19828638341361997584999276180, 3.04237992883956526382580180803, 4.02739006226397860422063723523, 5.17415559920020740235004505023, 6.02105941699806851978137666731, 6.77110709257129543149751729979, 7.61651198978984711071610610186, 8.595950331297591650079919400425, 9.581013158431873426745339860848, 10.56693033128841055051830949355