L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.349 − 1.69i)3-s + (−0.499 + 0.866i)4-s + 3.69·5-s + (−1.29 + 1.15i)6-s + (−1.40 − 2.24i)7-s + 0.999·8-s + (−2.75 + 1.18i)9-s + (−1.84 − 3.20i)10-s − 1.47·11-s + (1.64 + 0.545i)12-s + (−1.34 − 2.33i)13-s + (−1.23 + 2.33i)14-s + (−1.29 − 6.27i)15-s + (−0.5 − 0.866i)16-s + (3.28 + 5.69i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.201 − 0.979i)3-s + (−0.249 + 0.433i)4-s + 1.65·5-s + (−0.528 + 0.469i)6-s + (−0.531 − 0.847i)7-s + 0.353·8-s + (−0.918 + 0.395i)9-s + (−0.584 − 1.01i)10-s − 0.445·11-s + (0.474 + 0.157i)12-s + (−0.374 − 0.648i)13-s + (−0.331 + 0.624i)14-s + (−0.334 − 1.62i)15-s + (−0.125 − 0.216i)16-s + (0.797 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.613798 - 0.744304i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.613798 - 0.744304i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.349 + 1.69i)T \) |
| 7 | \( 1 + (1.40 + 2.24i)T \) |
good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + (1.34 + 2.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.28 - 5.69i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.444 - 0.769i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 + (-1.25 + 2.17i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.40 - 5.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.38 - 2.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.05 + 3.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.00618 + 0.0107i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.49 - 6.05i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.60 + 2.78i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.45 - 5.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.86 - 4.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.73 + 8.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + (6.03 + 10.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.72 + 9.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.23 + 3.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.43 - 7.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.58 - 11.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.02103497418727532640357225691, −12.40714722615674699551365346622, −10.65753437091663909238260793522, −10.27277851509740775506465707334, −8.999889670278148119572426798756, −7.67823653668736374535779847882, −6.47806916425287717756983414324, −5.37424803847900132745699196753, −2.96816187328029168737610720753, −1.43612196194275166626716796011,
2.69230425303319982947513031520, 5.08360387653250249905864693145, 5.68636237895206756386659361644, 6.87986371331111101315767090232, 8.846352245948556128457438628791, 9.504701238221960355706801512549, 10.05968500629716346319034192056, 11.37350463318408199483191858968, 12.83101482163249803133184390572, 13.94766192463054388064119898496