| L(s) = 1 | + (0.587 − 0.809i)2-s + (1.50 + 2.59i)3-s + (0.927 + 2.85i)4-s + (−3.88 − 5.35i)5-s + (2.98 + 0.303i)6-s + (−2.73 − 8.42i)7-s + (6.65 + 2.16i)8-s + (−4.44 + 7.82i)9-s − 6.61·10-s + (−10.8 − 1.76i)11-s + (−6 + 6.70i)12-s + (10.7 + 7.77i)13-s + (−8.42 − 2.73i)14-s + (8.01 − 18.1i)15-s + (−4.04 + 2.93i)16-s + (3.52 + 4.85i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.404i)2-s + (0.502 + 0.864i)3-s + (0.231 + 0.713i)4-s + (−0.777 − 1.07i)5-s + (0.497 + 0.0506i)6-s + (−0.390 − 1.20i)7-s + (0.832 + 0.270i)8-s + (−0.494 + 0.869i)9-s − 0.661·10-s + (−0.987 − 0.160i)11-s + (−0.5 + 0.559i)12-s + (0.823 + 0.598i)13-s + (−0.601 − 0.195i)14-s + (0.534 − 1.21i)15-s + (−0.252 + 0.183i)16-s + (0.207 + 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0753i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.21032 + 0.0456940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21032 + 0.0456940i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.50 - 2.59i)T \) |
| 11 | \( 1 + (10.8 + 1.76i)T \) |
| good | 2 | \( 1 + (-0.587 + 0.809i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (3.88 + 5.35i)T + (-7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (2.73 + 8.42i)T + (-39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (-10.7 - 7.77i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-3.52 - 4.85i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (-2.81 + 8.67i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 - 17.5iT - 529T^{2} \) |
| 29 | \( 1 + (-25.1 + 8.15i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-5.01 - 3.64i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (6.05 + 18.6i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (5.32 + 1.72i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 26.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-16.8 - 5.47i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (13.0 - 18.0i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-20.8 + 6.77i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-75.5 + 54.8i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + 76.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + (38.6 + 53.2i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-4.64 - 14.2i)T + (-4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (95.5 + 69.3i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (65.3 + 89.9i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 - 97.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-98.0 - 71.2i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−16.22369990015172662465551615964, −15.81047128871821161551034262406, −13.77995576401512308025984549244, −13.02924342535841602640677356415, −11.54205970484685680974067331237, −10.39848708828318557565026805875, −8.666182042785237578797577072728, −7.62818304482763887511519592706, −4.60554845048805868125646246277, −3.54700713091423210320113727816,
2.84455789384527626355174573627, 5.82311813473837448471582152982, 7.01324888295962943571367998969, 8.332120368689853386504063754768, 10.28525969745534632167204427188, 11.68450557581435557043828754577, 13.02273127162520003552343072347, 14.34430372356039070995757848842, 15.24297430347823786825356875044, 15.87074968266139324714277756836
