L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + i·5-s + (−1.24 + 0.719i)7-s − 0.999i·8-s + (0.5 − 0.866i)10-s + (−3.49 − 2.01i)11-s + (3.13 + 1.78i)13-s + 1.43·14-s + (−0.5 + 0.866i)16-s + (−1.15 − 1.99i)17-s + (−0.346 + 0.199i)19-s + (−0.866 + 0.499i)20-s + (2.01 + 3.49i)22-s + (−0.780 + 1.35i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (−0.471 + 0.272i)7-s − 0.353i·8-s + (0.158 − 0.273i)10-s + (−1.05 − 0.608i)11-s + (0.868 + 0.495i)13-s + 0.384·14-s + (−0.125 + 0.216i)16-s + (−0.279 − 0.484i)17-s + (−0.0794 + 0.0458i)19-s + (−0.193 + 0.111i)20-s + (0.430 + 0.745i)22-s + (−0.162 + 0.281i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03687500328\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03687500328\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-3.13 - 1.78i)T \) |
good | 7 | \( 1 + (1.24 - 0.719i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.49 + 2.01i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.15 + 1.99i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.346 - 0.199i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.780 - 1.35i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.93 - 6.81i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.10iT - 31T^{2} \) |
| 37 | \( 1 + (5.40 + 3.11i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.659 - 0.380i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.50 + 9.53i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.84iT - 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 1.73i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.84 - 4.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.00 + 5.20i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.19 - 3i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 0.931T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (-0.840 - 0.485i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.6 - 6.15i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.276814519900378759103798692380, −8.743434702272029545329991063798, −7.81234905837498976932963344734, −6.99184888202638060500197671277, −6.12564009023160983733291903018, −5.18857376685318555060431092749, −3.72725075031705035499679429649, −2.98486395523118411141830724109, −1.83438160712512085534375295665, −0.01883331932015040725184724232,
1.56294525018509141425314880919, 2.92062205822896297139152361149, 4.20040069329429066266596614755, 5.23166272392813952048219000696, 6.10874326657953401018963870777, 6.92023766988303010231252576950, 7.985801075731240600786617354533, 8.332204872732677722588278212058, 9.426958957470797227927925461314, 10.05798079356582246672412902178