| L(s) = 1 | + (0.313 + 1.37i)2-s + (1.51 + 1.01i)3-s + (−1.80 + 0.865i)4-s + (−0.618 − 3.10i)5-s + (−0.923 + 2.41i)6-s + (−1.63 − 0.325i)7-s + (−1.75 − 2.21i)8-s + (0.130 + 0.314i)9-s + (4.09 − 1.82i)10-s + (2.21 + 3.31i)11-s + (−3.61 − 0.516i)12-s + (−1.46 + 1.46i)13-s + (−0.0644 − 2.36i)14-s + (2.21 − 5.35i)15-s + (2.50 − 3.11i)16-s + (3.44 + 2.26i)17-s + ⋯ |
| L(s) = 1 | + (0.221 + 0.975i)2-s + (0.877 + 0.586i)3-s + (−0.901 + 0.432i)4-s + (−0.276 − 1.39i)5-s + (−0.377 + 0.985i)6-s + (−0.619 − 0.123i)7-s + (−0.621 − 0.783i)8-s + (0.0433 + 0.104i)9-s + (1.29 − 0.578i)10-s + (0.667 + 0.999i)11-s + (−1.04 − 0.149i)12-s + (−0.406 + 0.406i)13-s + (−0.0172 − 0.631i)14-s + (0.572 − 1.38i)15-s + (0.625 − 0.779i)16-s + (0.835 + 0.549i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.883919 + 0.610306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.883919 + 0.610306i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.313 - 1.37i)T \) |
| 17 | \( 1 + (-3.44 - 2.26i)T \) |
| good | 3 | \( 1 + (-1.51 - 1.01i)T + (1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (0.618 + 3.10i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (1.63 + 0.325i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-2.21 - 3.31i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (1.46 - 1.46i)T - 13iT^{2} \) |
| 19 | \( 1 + (5.50 + 2.28i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.448 + 0.299i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-3.34 + 0.664i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (5.34 - 7.99i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-1.82 + 2.72i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (1.27 - 6.40i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.07 + 1.27i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.17 - 1.17i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.80 + 6.77i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (4.92 + 11.8i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.19 - 0.435i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 + (-1.19 - 0.800i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-0.707 - 3.55i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-3.43 - 5.14i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.913 - 2.20i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-8.54 - 8.54i)T + 89iT^{2} \) |
| 97 | \( 1 + (6.56 - 1.30i)T + (89.6 - 37.1i)T^{2} \) |
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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
−14.99173043405357688832483214793, −14.27624955540774765853887193376, −12.84329910281972918378346250468, −12.31955979543069542224926550935, −9.799875576871796405463186808978, −9.068782681245002871908976736081, −8.217505860783629016770348497436, −6.66660868377369506816254657244, −4.83130007625612508348162808426, −3.83584066478321396449022622862,
2.60682156254395225048368742283, 3.55973374663253047400256849596, 6.08538143221590731161996445772, 7.62078578234942661441802992388, 8.924871005679687496764963296667, 10.25258251313872024316190689367, 11.22281874142303577174949905722, 12.42985251134736245680367812824, 13.56175426951265719361297123562, 14.35724280806328919706637012474
