| L(s) = 1 | + (−0.238 + 0.735i)2-s + (1.13 + 0.824i)4-s + (−0.877 + 1.51i)5-s + (−0.136 + 1.29i)7-s + (−2.12 + 1.54i)8-s + (−0.907 − 1.00i)10-s + (0.636 − 0.283i)11-s + (−2.94 − 0.626i)13-s + (−0.921 − 0.410i)14-s + (0.238 + 0.733i)16-s + (3.97 + 1.76i)17-s + (−3.31 + 0.703i)19-s + (−2.24 + 1.00i)20-s + (0.0563 + 0.536i)22-s + (3.64 − 2.64i)23-s + ⋯ |
| L(s) = 1 | + (−0.168 + 0.519i)2-s + (0.567 + 0.412i)4-s + (−0.392 + 0.679i)5-s + (−0.0515 + 0.490i)7-s + (−0.752 + 0.546i)8-s + (−0.287 − 0.318i)10-s + (0.192 − 0.0855i)11-s + (−0.817 − 0.173i)13-s + (−0.246 − 0.109i)14-s + (0.0595 + 0.183i)16-s + (0.964 + 0.429i)17-s + (−0.759 + 0.161i)19-s + (−0.502 + 0.223i)20-s + (0.0120 + 0.114i)22-s + (0.759 − 0.551i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.598350 + 0.992340i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.598350 + 0.992340i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 + (-5.26 - 1.81i)T \) |
| good | 2 | \( 1 + (0.238 - 0.735i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (0.877 - 1.51i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.136 - 1.29i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-0.636 + 0.283i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (2.94 + 0.626i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.97 - 1.76i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (3.31 - 0.703i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.64 + 2.64i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.14 - 3.52i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (5.28 + 9.14i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.08 - 3.42i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-5.10 + 1.08i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-1.51 - 4.65i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.43 + 13.6i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-3.12 + 3.46i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + (-2.43 + 4.21i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.47 + 14.0i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (9.16 - 4.07i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-10.6 - 4.75i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.0617 - 0.0686i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-5.20 - 3.78i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (1.91 + 1.38i)T + (29.9 + 92.2i)T^{2} \) |
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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
−12.23405302131214313238517506736, −11.22031400744904848735425020252, −10.44239345818614595592381754376, −9.110051255069203782605902895633, −8.142803992384711792119116112680, −7.26425148428372810106070074761, −6.47389148972993490240313015847, −5.32782548319478305782276200890, −3.57140657212132937052334839885, −2.46851435380065981953760915865,
0.959792270048012751192647275816, 2.65539206862184963127898148107, 4.16205965714675489223724131198, 5.39018853527476298764480853711, 6.70660760032591380225322981291, 7.62002015095620355776146067248, 8.875568524086983758872117664956, 9.856821892361385506336557215618, 10.55084382532809565371147170379, 11.73705093621654407402901110115
