L(s) = 1 | − 2.73i·2-s + (0.280 + 1.70i)3-s − 5.49·4-s − 2.40·5-s + (4.68 − 0.768i)6-s + (−1.28 + 2.31i)7-s + 9.58i·8-s + (−2.84 + 0.959i)9-s + 6.58i·10-s − i·11-s + (−1.54 − 9.39i)12-s − 1.84i·13-s + (6.33 + 3.51i)14-s + (−0.675 − 4.11i)15-s + 15.2·16-s − 5.51·17-s + ⋯ |
L(s) = 1 | − 1.93i·2-s + (0.162 + 0.986i)3-s − 2.74·4-s − 1.07·5-s + (1.91 − 0.313i)6-s + (−0.485 + 0.874i)7-s + 3.38i·8-s + (−0.947 + 0.319i)9-s + 2.08i·10-s − 0.301i·11-s + (−0.445 − 2.71i)12-s − 0.511i·13-s + (1.69 + 0.940i)14-s + (−0.174 − 1.06i)15-s + 3.80·16-s − 1.33·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.199033 + 0.140100i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.199033 + 0.140100i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.280 - 1.70i)T \) |
| 7 | \( 1 + (1.28 - 2.31i)T \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 + 2.73iT - 2T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 13 | \( 1 + 1.84iT - 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 - 4.50iT - 19T^{2} \) |
| 23 | \( 1 + 0.776iT - 23T^{2} \) |
| 29 | \( 1 + 2.13iT - 29T^{2} \) |
| 31 | \( 1 - 1.00iT - 31T^{2} \) |
| 37 | \( 1 - 0.563T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 - 4.25T + 47T^{2} \) |
| 53 | \( 1 - 6.96iT - 53T^{2} \) |
| 59 | \( 1 + 1.34T + 59T^{2} \) |
| 61 | \( 1 - 11.1iT - 61T^{2} \) |
| 67 | \( 1 - 6.52T + 67T^{2} \) |
| 71 | \( 1 - 5.51iT - 71T^{2} \) |
| 73 | \( 1 - 6.97iT - 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 1.17T + 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−12.02921333926478109801078490895, −11.42731953788190094661788369813, −10.60809390723822619137151703951, −9.755527045111456635133953084392, −8.802490163643443275061422796248, −8.214019189550588179846261863272, −5.63118105500907865161270939683, −4.39165014088491437603042429611, −3.57235203619227949710076440940, −2.55307689973799021227253838592,
0.19382595912045357871954363861, 3.76104378946074672142992294690, 4.83980555559091859731236768195, 6.47785454058569629303377026747, 6.95919144142917300926565843619, 7.69467844931897912148975587297, 8.575385455232001740668694105987, 9.472369876476216615594122991937, 11.16641091321130549544378791115, 12.47984348526173375449971419178