| L(s) = 1 | + (1.19 + 0.813i)2-s + (1.71 − 0.210i)3-s + (0.0320 + 0.0815i)4-s + (−0.000507 + 0.00222i)5-s + (2.22 + 1.14i)6-s + (0.314 − 2.62i)7-s + (0.614 − 2.69i)8-s + (2.91 − 0.725i)9-s + (−0.00241 + 0.00223i)10-s + (−2.93 + 1.41i)11-s + (0.0722 + 0.133i)12-s + (1.97 + 1.34i)13-s + (2.51 − 2.88i)14-s + (−0.000403 + 0.00392i)15-s + (3.05 − 2.83i)16-s + (0.0598 − 0.152i)17-s + ⋯ |
| L(s) = 1 | + (0.844 + 0.575i)2-s + (0.992 − 0.121i)3-s + (0.0160 + 0.0407i)4-s + (−0.000226 + 0.000993i)5-s + (0.907 + 0.468i)6-s + (0.118 − 0.992i)7-s + (0.217 − 0.952i)8-s + (0.970 − 0.241i)9-s + (−0.000763 + 0.000708i)10-s + (−0.884 + 0.425i)11-s + (0.0208 + 0.0385i)12-s + (0.546 + 0.372i)13-s + (0.671 − 0.769i)14-s + (−0.000104 + 0.00101i)15-s + (0.763 − 0.708i)16-s + (0.0145 − 0.0369i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.79762 - 0.0117790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.79762 - 0.0117790i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 + 0.210i)T \) |
| 7 | \( 1 + (-0.314 + 2.62i)T \) |
| good | 2 | \( 1 + (-1.19 - 0.813i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (0.000507 - 0.00222i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (2.93 - 1.41i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (-1.97 - 1.34i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.0598 + 0.152i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (1.97 - 3.42i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.65 - 5.83i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (3.91 - 0.589i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (2.86 - 4.96i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.85 + 0.882i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (-2.43 + 0.750i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (2.71 + 0.836i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-0.462 - 0.315i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (8.34 + 1.25i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-4.53 - 1.39i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-5.47 + 13.9i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (2.35 - 4.07i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.98 - 6.25i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (0.543 + 7.24i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (-6.21 - 10.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-13.3 + 9.08i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (6.52 - 4.44i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (-3.00 + 5.20i)T + (-48.5 - 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.97382435243125854241210030448, −10.11101818462718024870098935928, −9.381775398215754935433556554315, −8.061100101424918135426368680152, −7.38769155056640560723257076458, −6.53269485417910468744953759329, −5.27315745862129968827779991655, −4.19008088941263594266512835086, −3.45284677515768710478987857766, −1.60602509462789368011036820447,
2.27891319545199188336160611014, 2.88879412407809924508597793019, 4.09125756483153066134175366350, 5.05560859742995612622852506480, 6.17733322398346711377549901779, 7.82137303544940010872546796334, 8.390713304704661427229754882725, 9.159043246707449054052916397511, 10.44085801894754140861987067764, 11.18124971318582447440446111823
