| L(s) = 1 | + (0.142 + 0.989i)2-s + (0.909 + 0.415i)3-s + (−0.959 + 0.281i)4-s + (−2.10 + 2.43i)5-s + (−0.281 + 0.959i)6-s + (−2.62 + 0.359i)7-s + (−0.415 − 0.909i)8-s + (0.654 + 0.755i)9-s + (−2.70 − 1.74i)10-s + (−0.264 − 0.0380i)11-s + (−0.989 − 0.142i)12-s + (−1.37 + 2.14i)13-s + (−0.728 − 2.54i)14-s + (−2.93 + 1.33i)15-s + (0.841 − 0.540i)16-s + (−1.86 − 0.548i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (0.525 + 0.239i)3-s + (−0.479 + 0.140i)4-s + (−0.943 + 1.08i)5-s + (−0.115 + 0.391i)6-s + (−0.990 + 0.135i)7-s + (−0.146 − 0.321i)8-s + (0.218 + 0.251i)9-s + (−0.856 − 0.550i)10-s + (−0.0798 − 0.0114i)11-s + (−0.285 − 0.0410i)12-s + (−0.382 + 0.595i)13-s + (−0.194 − 0.679i)14-s + (−0.756 + 0.345i)15-s + (0.210 − 0.135i)16-s + (−0.453 − 0.133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.215721 - 0.324168i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.215721 - 0.324168i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (2.62 - 0.359i)T \) |
| 23 | \( 1 + (1.46 + 4.56i)T \) |
| good | 5 | \( 1 + (2.10 - 2.43i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.264 + 0.0380i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.37 - 2.14i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.86 + 0.548i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-7.10 + 2.08i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (8.36 + 2.45i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.77 - 1.26i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.69 + 2.33i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (8.31 + 7.20i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.76 + 1.26i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + (-2.84 - 4.42i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (3.09 - 4.82i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (4.46 + 9.77i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (9.51 - 1.36i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.86 - 12.9i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.36 - 11.4i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.44 + 3.80i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (4.64 + 5.35i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (4.57 - 10.0i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-0.645 + 0.744i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.46407804015289705087651789361, −9.533849071322800236758516635315, −8.996743402388984275506116417386, −7.81415737074960393374335211204, −7.20832247724878105141806920464, −6.64828472845842372850214038473, −5.49895254096727581334517022019, −4.23330133495000665124076787142, −3.46381087635932820153739739554, −2.60826957940930476725694694309,
0.16274377905644184404492563482, 1.57223935238432323982073934993, 3.21064615465868721734422021313, 3.68664016758985477589802698001, 4.85809767127901150117298288183, 5.75731000139656592872376452406, 7.22070945726106784078969655013, 7.85403185749231468263518194562, 8.731157238130410470172362874310, 9.517201987697705386559562364409
