L(s) = 1 | + (0.533 − 1.30i)2-s + (1.59 − 0.667i)3-s + (−1.43 − 1.39i)4-s + (0.143 + 2.23i)5-s + (−0.0218 − 2.44i)6-s + (0.582 − 0.582i)7-s + (−2.59 + 1.13i)8-s + (2.10 − 2.13i)9-s + (2.99 + 1.00i)10-s − 3.68·11-s + (−3.22 − 1.27i)12-s + (−3.88 + 3.88i)13-s + (−0.452 − 1.07i)14-s + (1.71 + 3.47i)15-s + (0.0980 + 3.99i)16-s + (−0.880 − 0.880i)17-s + ⋯ |
L(s) = 1 | + (0.377 − 0.926i)2-s + (0.922 − 0.385i)3-s + (−0.715 − 0.698i)4-s + (0.0639 + 0.997i)5-s + (−0.00892 − 0.999i)6-s + (0.220 − 0.220i)7-s + (−0.916 + 0.399i)8-s + (0.703 − 0.711i)9-s + (0.948 + 0.316i)10-s − 1.11·11-s + (−0.929 − 0.368i)12-s + (−1.07 + 1.07i)13-s + (−0.120 − 0.287i)14-s + (0.443 + 0.896i)15-s + (0.0245 + 0.999i)16-s + (−0.213 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18738 - 0.860090i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18738 - 0.860090i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.533 + 1.30i)T \) |
| 3 | \( 1 + (-1.59 + 0.667i)T \) |
| 5 | \( 1 + (-0.143 - 2.23i)T \) |
good | 7 | \( 1 + (-0.582 + 0.582i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 + (3.88 - 3.88i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.880 + 0.880i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 + (-2.06 + 2.06i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.37iT - 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 + (2.44 + 2.44i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.648iT - 41T^{2} \) |
| 43 | \( 1 + (0.819 - 0.819i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.28 + 6.28i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.60 + 5.60i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.12iT - 59T^{2} \) |
| 61 | \( 1 + 5.13iT - 61T^{2} \) |
| 67 | \( 1 + (4.90 + 4.90i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.13iT - 71T^{2} \) |
| 73 | \( 1 + (-4.69 - 4.69i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.10iT - 79T^{2} \) |
| 83 | \( 1 + (-6.27 - 6.27i)T + 83iT^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + (5.42 - 5.42i)T - 97iT^{2} \) |
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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
−13.43137421447527098548799335588, −12.28420714829393539449789436494, −11.28000860943481382103855553457, −10.12445798610275017196168575639, −9.367623988962739086909988371636, −7.85855487078356810439648631930, −6.76419059555456871234573654010, −4.89356260491190917983625212336, −3.28466127136667913066239566178, −2.22467802234956333354550324128,
3.01911731985750245350868598566, 4.75019337651072685324656060181, 5.42216905625436586800682869813, 7.55028747711155399544709683977, 8.099143745639525942010466027594, 9.209465464448066445009749897079, 10.11774493687689452953203006867, 12.08267538006752368062601360812, 13.07465919548619245949263750474, 13.64813866771310253925024524517