| L(s) = 1 | + (−0.129 − 0.0359i)2-s + (−1.73 − 0.0555i)3-s + (−1.69 − 1.02i)4-s + (−2.51 + 0.0976i)5-s + (0.221 + 0.0694i)6-s + (4.08 + 0.638i)7-s + (0.366 + 0.388i)8-s + (2.99 + 0.192i)9-s + (0.328 + 0.0778i)10-s + (0.654 + 4.78i)11-s + (2.88 + 1.86i)12-s + (−0.189 + 0.276i)13-s + (−0.504 − 0.229i)14-s + (4.36 − 0.0292i)15-s + (1.81 + 3.44i)16-s + (1.56 − 0.787i)17-s + ⋯ |
| L(s) = 1 | + (−0.0913 − 0.0254i)2-s + (−0.999 − 0.0320i)3-s + (−0.848 − 0.512i)4-s + (−1.12 + 0.0436i)5-s + (0.0904 + 0.0283i)6-s + (1.54 + 0.241i)7-s + (0.129 + 0.137i)8-s + (0.997 + 0.0641i)9-s + (0.103 + 0.0246i)10-s + (0.197 + 1.44i)11-s + (0.831 + 0.539i)12-s + (−0.0526 + 0.0767i)13-s + (−0.134 − 0.0613i)14-s + (1.12 − 0.00753i)15-s + (0.453 + 0.860i)16-s + (0.380 − 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.575132 + 0.261831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.575132 + 0.261831i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.73 + 0.0555i)T \) |
| good | 2 | \( 1 + (0.129 + 0.0359i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (2.51 - 0.0976i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-4.08 - 0.638i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.654 - 4.78i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (0.189 - 0.276i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.56 + 0.787i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.123 - 2.12i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (0.620 - 1.60i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (0.238 + 0.170i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-5.27 + 6.04i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-6.18 - 8.31i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (1.90 - 7.40i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (0.684 + 0.620i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-4.38 - 5.02i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (2.13 - 12.1i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (1.52 - 0.623i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (0.475 - 0.287i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-6.22 + 4.45i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (3.03 + 10.1i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (10.8 - 2.56i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (11.3 - 11.1i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (3.20 + 12.4i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-2.10 + 7.03i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-0.615 - 0.0238i)T + (96.7 + 7.51i)T^{2} \) |
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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
−11.86534383283567414417030454733, −11.55028349220291274113933317321, −10.39361262185237025482087078019, −9.534345430322878467188276083136, −8.081757587106059597080575176354, −7.51216901992371893947062525048, −5.93450892877951241394853305219, −4.60937686813375104749206449417, −4.43630537475269002817815703782, −1.42152820978122568087553991287,
0.71902433948807723552763746814, 3.73053114554026846605810967299, 4.58809618743614698751697555267, 5.55449244988908743283175404356, 7.21419585375499787002651018214, 8.100540926222302267958369772527, 8.736777399157592719900552812607, 10.34275204862052119923466445015, 11.30774395671165202387670498076, 11.72928629350791536303077295452
