L(s) = 1 | + (−1.41 + 0.0786i)2-s + (0.637 − 1.61i)3-s + (1.98 − 0.222i)4-s + (0.686 + 0.396i)5-s + (−0.774 + 2.32i)6-s + (−2.35 + 1.35i)7-s + (−2.78 + 0.469i)8-s + (−2.18 − 2.05i)9-s + (−0.999 − 0.505i)10-s + (1.71 + 2.96i)11-s + (0.910 − 3.34i)12-s + (−1.68 + 2.92i)13-s + (3.21 − 2.10i)14-s + (1.07 − 0.852i)15-s + (3.90 − 0.882i)16-s − 2.52i·17-s + ⋯ |
L(s) = 1 | + (−0.998 + 0.0556i)2-s + (0.368 − 0.929i)3-s + (0.993 − 0.111i)4-s + (0.306 + 0.177i)5-s + (−0.316 + 0.948i)6-s + (−0.888 + 0.513i)7-s + (−0.986 + 0.166i)8-s + (−0.728 − 0.684i)9-s + (−0.316 − 0.159i)10-s + (0.516 + 0.894i)11-s + (0.262 − 0.964i)12-s + (−0.467 + 0.809i)13-s + (0.858 − 0.561i)14-s + (0.277 − 0.220i)15-s + (0.975 − 0.220i)16-s − 0.612i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.539566 - 0.128772i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.539566 - 0.128772i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0786i)T \) |
| 3 | \( 1 + (-0.637 + 1.61i)T \) |
good | 5 | \( 1 + (-0.686 - 0.396i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.35 - 1.35i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.71 - 2.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.68 - 2.92i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2.52iT - 17T^{2} \) |
| 19 | \( 1 + 2.20iT - 19T^{2} \) |
| 23 | \( 1 + (-1.07 + 1.86i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.686 - 0.396i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.47 - 0.852i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + (-0.127 - 0.0737i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.01 + 3.47i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.77 + 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.51iT - 53T^{2} \) |
| 59 | \( 1 + (2.58 - 4.48i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.68 + 2.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.01 - 3.47i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 + 2.37T + 73T^{2} \) |
| 79 | \( 1 + (8.80 - 5.08i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.62 - 6.28i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.34iT - 89T^{2} \) |
| 97 | \( 1 + (-6.24 - 10.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−16.73210382029695968105843262199, −15.36727123283390770636544800957, −14.18010902533107633907303837178, −12.60065656763258940379422092608, −11.68604662922207711970596641855, −9.784983529081480783211548529942, −8.930884703281486522628767796269, −7.25147504858081772664063154981, −6.34259575125884772498092378583, −2.44171528296503042666414328994,
3.35888230345847000471252667227, 6.00136149408027008257477790444, 7.927550193705076381336876952662, 9.289588685598248005923836495337, 10.10775794044941145069587435610, 11.24510184214291654017450503633, 13.02102111206209898506286307331, 14.57156996359569565933870937295, 15.78783725616046755500188532613, 16.66969579784214730498949882954