L(s) = 1 | + (0.430 + 1.60i)2-s + (−1.35 − 1.08i)3-s + (−0.661 + 0.382i)4-s + (1.15 − 2.63i)6-s + (1.73 − 0.465i)7-s + (1.45 + 1.45i)8-s + (0.661 + 2.92i)9-s + (3.12 + 1.80i)11-s + (1.30 + 0.198i)12-s + (1.27 + 0.342i)13-s + (1.49 + 2.59i)14-s + (−2.47 + 4.28i)16-s + (−0.277 + 0.277i)17-s + (−4.41 + 2.32i)18-s − 6.25i·19-s + ⋯ |
L(s) = 1 | + (0.304 + 1.13i)2-s + (−0.781 − 0.624i)3-s + (−0.330 + 0.191i)4-s + (0.471 − 1.07i)6-s + (0.656 − 0.175i)7-s + (0.513 + 0.513i)8-s + (0.220 + 0.975i)9-s + (0.942 + 0.544i)11-s + (0.377 + 0.0573i)12-s + (0.354 + 0.0950i)13-s + (0.399 + 0.692i)14-s + (−0.618 + 1.07i)16-s + (−0.0671 + 0.0671i)17-s + (−1.04 + 0.547i)18-s − 1.43i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14866 + 0.677021i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14866 + 0.677021i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.35 + 1.08i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.430 - 1.60i)T + (-1.73 + i)T^{2} \) |
| 7 | \( 1 + (-1.73 + 0.465i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.12 - 1.80i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 0.342i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (0.277 - 0.277i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.25iT - 19T^{2} \) |
| 23 | \( 1 + (0.579 - 2.16i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (1.56 - 2.71i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.55 + 5.55i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.29 + 0.744i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.10 - 4.10i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.02 - 3.82i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (7.48 + 7.48i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.279 - 0.483i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.96 - 5.13i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.90 + 10.8i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 8.01iT - 71T^{2} \) |
| 73 | \( 1 + (-1.29 + 1.29i)T - 73iT^{2} \) |
| 79 | \( 1 + (-6.96 - 4.02i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.560 + 0.150i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + (5.14 - 1.37i)T + (84.0 - 48.5i)T^{2} \) |
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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
−12.51539940678643916874293231548, −11.33483560638339040570898905246, −10.91603931469000032950664290422, −9.253628896875059394424749632883, −7.969339990791256169179567053072, −7.13751853537915012956413231277, −6.42359079603810495241911182587, −5.33462363762947447986614115786, −4.40638716128401856462819674694, −1.74383272779161177283296441205,
1.46803322556119830886126390119, 3.43540110291468377881803292703, 4.32257269703169355042133581743, 5.59945240510160999583513450912, 6.78168517371947728180189166795, 8.370207661895989345650639794896, 9.583721395123090415084349231528, 10.50439158043150642003764164403, 11.20155544246882622008451858405, 11.93241141009451060736681605210