| L(s) = 1 | + (0.0883 + 1.41i)2-s + (−0.909 − 0.415i)3-s + (−1.98 + 0.249i)4-s + (2.58 + 2.24i)5-s + (0.506 − 1.32i)6-s + (2.40 − 1.54i)7-s + (−0.527 − 2.77i)8-s + (0.654 + 0.755i)9-s + (−2.93 + 3.85i)10-s + (−0.649 + 4.51i)11-s + (1.90 + 0.597i)12-s + (−0.0740 − 0.0475i)13-s + (2.39 + 3.25i)14-s + (−1.42 − 3.11i)15-s + (3.87 − 0.989i)16-s + (−0.829 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (0.0624 + 0.998i)2-s + (−0.525 − 0.239i)3-s + (−0.992 + 0.124i)4-s + (1.15 + 1.00i)5-s + (0.206 − 0.539i)6-s + (0.908 − 0.584i)7-s + (−0.186 − 0.982i)8-s + (0.218 + 0.251i)9-s + (−0.929 + 1.21i)10-s + (−0.195 + 1.36i)11-s + (0.550 + 0.172i)12-s + (−0.0205 − 0.0131i)13-s + (0.639 + 0.870i)14-s + (−0.367 − 0.804i)15-s + (0.968 − 0.247i)16-s + (−0.201 + 0.684i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.752765 + 1.00499i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.752765 + 1.00499i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0883 - 1.41i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 23 | \( 1 + (-4.59 + 1.36i)T \) |
| good | 5 | \( 1 + (-2.58 - 2.24i)T + (0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-2.40 + 1.54i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.649 - 4.51i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (0.0740 + 0.0475i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.829 - 2.82i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (5.54 - 1.62i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.509 + 0.149i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.376 - 0.172i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.57 + 2.23i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-8.21 + 9.47i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.324 - 0.711i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 5.43iT - 47T^{2} \) |
| 53 | \( 1 + (4.65 + 7.24i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (3.13 - 4.87i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.80 + 2.19i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.75 + 12.1i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-9.99 + 1.43i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-5.96 + 1.75i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (11.2 + 7.25i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-8.61 - 9.94i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (5.30 + 2.42i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (11.1 + 9.63i)T + (13.8 + 96.0i)T^{2} \) |
| 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
−12.50229884445284813460145224108, −10.85455315834738484689449279835, −10.37619499915501582436423186640, −9.325688509669836519402849454204, −7.961836321750777717480332335189, −7.03483031864803337229972228455, −6.37740791748761028902029408092, −5.28332826201358899205447047259, −4.22676370906929210122897892556, −1.99402185802950186034847221436,
1.15601499610235024503288925681, 2.61682593599144349638248205448, 4.52808010464546330915471310099, 5.27974607464215203436714866736, 6.06820736523030112998663375361, 8.277818223853255943191540663759, 8.971487341642004421896143091729, 9.706300810992645736447860482362, 11.00626475818188814218495427633, 11.32677204961586208417846648656
