| L(s) = 1 | + (0.866 + 0.5i)2-s + (−1.55 + 0.762i)3-s + (0.499 + 0.866i)4-s + 5-s + (−1.72 − 0.117i)6-s + (1.13 − 2.38i)7-s + 0.999i·8-s + (1.83 − 2.37i)9-s + (0.866 + 0.5i)10-s + 1.06i·11-s + (−1.43 − 0.965i)12-s + (3.43 + 1.98i)13-s + (2.18 − 1.49i)14-s + (−1.55 + 0.762i)15-s + (−0.5 + 0.866i)16-s + (3.26 − 5.65i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.897 + 0.440i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.705 − 0.0478i)6-s + (0.430 − 0.902i)7-s + 0.353i·8-s + (0.612 − 0.790i)9-s + (0.273 + 0.158i)10-s + 0.321i·11-s + (−0.415 − 0.278i)12-s + (0.952 + 0.549i)13-s + (0.582 − 0.400i)14-s + (−0.401 + 0.196i)15-s + (−0.125 + 0.216i)16-s + (0.792 − 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75088 + 0.707032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75088 + 0.707032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (1.55 - 0.762i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1.13 + 2.38i)T \) |
| good | 11 | \( 1 - 1.06iT - 11T^{2} \) |
| 13 | \( 1 + (-3.43 - 1.98i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.26 + 5.65i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.00 + 0.582i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.37iT - 23T^{2} \) |
| 29 | \( 1 + (5.89 - 3.40i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.61 + 2.08i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.89 - 5.02i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.22 + 9.05i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.665 - 1.15i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.02 - 10.4i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.55 - 4.36i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.63 + 8.02i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.02 - 1.74i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.98 + 5.16i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.707iT - 71T^{2} \) |
| 73 | \( 1 + (6.25 + 3.60i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.43 + 5.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.68 - 4.65i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.09 + 3.62i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.59 + 2.65i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.98219845009374291997904204186, −9.855586692171071441806266365688, −9.219681484238433432803989597351, −7.66961151628452815278631950891, −7.04437440421733603496833895392, −6.01260492885335918924305708459, −5.21662582470006154043854296697, −4.36147483554733890441249690730, −3.37388752636170807478908829924, −1.33119998171337695791379669793,
1.26315747729538999635852736770, 2.46606679477854672502896477386, 3.96520029181593758981499761776, 5.21763503119587692724245340520, 5.89714306043085462229419252189, 6.39969155075561451305082741571, 7.86449275694791251453298785396, 8.648446997313151599314474650802, 10.00935593567874782632444616315, 10.69236428370259869501855583413
