| L(s) = 1 | + (0.325 + 1.37i)2-s + (0.281 + 1.86i)3-s + (−1.78 + 0.895i)4-s + (−2.47 − 0.970i)5-s + (−2.48 + 0.995i)6-s + (−2.31 − 1.28i)7-s + (−1.81 − 2.17i)8-s + (−0.546 + 0.168i)9-s + (0.531 − 3.71i)10-s + (−0.464 + 1.50i)11-s + (−2.17 − 3.09i)12-s + (−0.149 + 0.0341i)13-s + (1.02 − 3.59i)14-s + (1.11 − 4.89i)15-s + (2.39 − 3.20i)16-s + (0.505 − 0.344i)17-s + ⋯ |
| L(s) = 1 | + (0.229 + 0.973i)2-s + (0.162 + 1.07i)3-s + (−0.894 + 0.447i)4-s + (−1.10 − 0.433i)5-s + (−1.01 + 0.406i)6-s + (−0.873 − 0.487i)7-s + (−0.641 − 0.767i)8-s + (−0.182 + 0.0561i)9-s + (0.168 − 1.17i)10-s + (−0.140 + 0.454i)11-s + (−0.628 − 0.892i)12-s + (−0.0414 + 0.00945i)13-s + (0.273 − 0.961i)14-s + (0.288 − 1.26i)15-s + (0.599 − 0.800i)16-s + (0.122 − 0.0836i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.200761 - 0.266409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.200761 - 0.266409i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.325 - 1.37i)T \) |
| 7 | \( 1 + (2.31 + 1.28i)T \) |
| good | 3 | \( 1 + (-0.281 - 1.86i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (2.47 + 0.970i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (0.464 - 1.50i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (0.149 - 0.0341i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.505 + 0.344i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (6.85 - 3.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.07 + 1.41i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (1.12 - 2.33i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (2.17 - 3.76i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.622 - 0.0466i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (5.51 + 6.91i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-3.95 - 3.15i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (4.42 - 4.10i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.0118 + 0.000887i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-4.81 + 1.89i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (8.84 - 0.662i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (12.0 + 6.98i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.8 + 5.21i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-4.24 - 3.94i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-3.68 - 6.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.36 - 0.767i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (6.79 - 2.09i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 1.13T + 97T^{2} \) |
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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
−12.43205239470726461928892227414, −10.75544821734196510430241161662, −9.982438911444112691267251912000, −9.090891845055527747621670479742, −8.242720997774975426516374252437, −7.30845498755378070728714775840, −6.31075626484157855865448456517, −4.91100289488103112796579319364, −4.09947632695889745571394415644, −3.52565263332456583665630963330,
0.19477950429901587649478139805, 2.16457504676938121753678369077, 3.23829329962572050802132210891, 4.31618292959167958215511829259, 5.92745383301740868507567807321, 6.87237518829333711644698141836, 7.979983696485231546459977009615, 8.761234419062499762889776974811, 9.915668922250685358518394870589, 10.94523751689296576401544247026
