L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−2 + i)5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (1.23 − 1.86i)10-s + (−0.5 − 0.866i)11-s + (−0.866 + 3.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s + (−0.5 + 0.866i)19-s + (−0.133 + 2.23i)20-s + (0.866 + 0.499i)22-s + (3 − 4i)25-s + (−1 − 3.46i)26-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.894 + 0.447i)5-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (0.389 − 0.590i)10-s + (−0.150 − 0.261i)11-s + (−0.240 + 0.970i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s + (−0.114 + 0.198i)19-s + (−0.0299 + 0.499i)20-s + (0.184 + 0.106i)22-s + (0.600 − 0.800i)25-s + (−0.196 − 0.679i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0342 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0342 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3714607738\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3714607738\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 - i)T \) |
| 13 | \( 1 + (0.866 - 3.5i)T \) |
good | 7 | \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5 + 8.66i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.46 + 2i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 18iT - 83T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.19 + 3i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.640506718186804802510785630443, −8.604187964304632515114220516267, −7.910007437480326906329489892784, −7.19446850166932984574831311106, −6.45987577433407175190709726417, −5.50568523150304171987041542319, −4.21658478025145559032394915832, −3.39718233709410254065277308353, −1.99185280347326619741024644373, −0.21698004197574195217583510936,
1.21151505400892189639481573550, 2.83873652378764366855240172030, 3.59494008506650607611055993655, 4.81646836085032070474291628042, 5.68306981784099055458418441048, 7.06808493312154392678169947435, 7.60057434375168937823862801493, 8.413074145691480955418187166445, 9.159379705842173871845774792305, 9.939029815311853982437322999942