L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.05 + 1.83i)5-s + 0.999·6-s + (−1.11 − 2.39i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.05 + 1.83i)10-s + (−1.67 − 2.90i)11-s + (0.499 − 0.866i)12-s − 4.23·13-s + (−2.63 − 0.231i)14-s − 2.11·15-s + (−0.5 + 0.866i)16-s + (−3.47 − 6.02i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.473 + 0.820i)5-s + 0.408·6-s + (−0.422 − 0.906i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.334 + 0.580i)10-s + (−0.505 − 0.875i)11-s + (0.144 − 0.249i)12-s − 1.17·13-s + (−0.704 − 0.0619i)14-s − 0.546·15-s + (−0.125 + 0.216i)16-s + (−0.843 − 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.189267 - 0.742430i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.189267 - 0.742430i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.11 + 2.39i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (1.05 - 1.83i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.67 + 2.90i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + (3.47 + 6.02i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.59 + 6.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.29T + 29T^{2} \) |
| 31 | \( 1 + (2.27 + 3.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.126 + 0.218i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + (3.57 - 6.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.878 + 1.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.52 + 11.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.45 - 5.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.15 - 3.72i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + (-6.38 - 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.13 - 1.96i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + (1.89 - 3.28i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
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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
−10.09706828266556924218147883891, −9.352522325166299739428565885186, −8.266879638054880949791702768327, −7.22025897467614914382949613941, −6.54582492670127599472189518226, −5.05702266793021028543682566457, −4.34695367096310123993979429012, −3.14230253370120729577936608194, −2.68118609878816025019581372597, −0.30720162304670363033147764612,
1.97606054308563600801443743768, 3.20757446524576624713459809430, 4.58545504901171073801109736671, 5.20106599863677353741189142792, 6.36920321861247214416480843157, 7.16713133365417573396281182373, 8.070803482241944853841641297711, 8.730968900077356071820552843764, 9.450000076511385056805963691009, 10.54193575443870570066538139281