| L(s) = 1 | + (0.0182 + 0.0465i)2-s + (0.00478 + 1.73i)3-s + (1.46 − 1.35i)4-s + (3.32 + 1.59i)5-s + (−0.0805 + 0.0318i)6-s + (1.96 + 1.76i)7-s + (0.180 + 0.0867i)8-s + (−2.99 + 0.0165i)9-s + (−0.0137 + 0.183i)10-s + (−0.0359 + 0.0450i)11-s + (2.36 + 2.52i)12-s + (−0.421 − 1.07i)13-s + (−0.0463 + 0.123i)14-s + (−2.75 + 5.76i)15-s + (0.297 − 3.97i)16-s + (−4.28 − 3.97i)17-s + ⋯ |
| L(s) = 1 | + (0.0129 + 0.0329i)2-s + (0.00276 + 0.999i)3-s + (0.732 − 0.679i)4-s + (1.48 + 0.715i)5-s + (−0.0328 + 0.0130i)6-s + (0.743 + 0.668i)7-s + (0.0636 + 0.0306i)8-s + (−0.999 + 0.00552i)9-s + (−0.00435 + 0.0581i)10-s + (−0.0108 + 0.0135i)11-s + (0.681 + 0.730i)12-s + (−0.116 − 0.297i)13-s + (−0.0124 + 0.0331i)14-s + (−0.711 + 1.48i)15-s + (0.0744 − 0.993i)16-s + (−1.03 − 0.963i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.84982 + 0.900219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84982 + 0.900219i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.00478 - 1.73i)T \) |
| 7 | \( 1 + (-1.96 - 1.76i)T \) |
| good | 2 | \( 1 + (-0.0182 - 0.0465i)T + (-1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-3.32 - 1.59i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (0.0359 - 0.0450i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.421 + 1.07i)T + (-9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (4.28 + 3.97i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.80 + 6.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.40 - 6.15i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (3.42 - 1.05i)T + (23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (3.17 + 5.50i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.11 - 2.19i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-1.99 + 1.35i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (-5.78 - 3.94i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (-3.09 - 7.87i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-0.957 - 0.295i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-9.58 - 6.53i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (8.17 + 7.58i)T + (4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (2.65 + 4.60i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.721 + 3.16i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.401 - 0.0604i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (1.71 - 2.97i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.73 + 12.0i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (0.656 - 1.67i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (3.04 + 5.27i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.02036297988288568391739686261, −10.52072161328493631878930513288, −9.293017589758925132563091040576, −9.199263889462594329183532773828, −7.37709179914414862995675615257, −6.29787785079429426366357140923, −5.54172156710172812360208138843, −4.81864145200843907690769456247, −2.79595404627349476891211580725, −2.10256209005174510258807949494,
1.65644027149850988999247163052, 2.18525853731779033831102702556, 4.05981796265497015037304506593, 5.52954214670722399194549928060, 6.41413265099488556055613417741, 7.17071345669128596347189066376, 8.395543271240374486661734527073, 8.750208989351528874500693587013, 10.39061445925739066030550809647, 10.91368564883405442782392328636
