| L(s) = 1 | + (0.142 + 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (1.62 − 1.87i)5-s + (0.959 + 0.281i)6-s + (−0.841 + 0.540i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (2.09 + 1.34i)10-s + (0.670 − 4.66i)11-s + (−0.142 + 0.989i)12-s + (−4.04 − 2.59i)13-s + (−0.654 − 0.755i)14-s + (−1.03 − 2.26i)15-s + (0.841 − 0.540i)16-s + (−1.09 − 0.322i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.727 − 0.839i)5-s + (0.391 + 0.115i)6-s + (−0.317 + 0.204i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (0.661 + 0.424i)10-s + (0.202 − 1.40i)11-s + (−0.0410 + 0.285i)12-s + (−1.12 − 0.720i)13-s + (−0.175 − 0.201i)14-s + (−0.266 − 0.583i)15-s + (0.210 − 0.135i)16-s + (−0.266 − 0.0782i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0138 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0138 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.953402 - 0.940307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.953402 - 0.940307i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (4.77 + 0.441i)T \) |
| good | 5 | \( 1 + (-1.62 + 1.87i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.670 + 4.66i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (4.04 + 2.59i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.09 + 0.322i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (2.82 - 0.829i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (2.06 + 0.606i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.465 - 1.01i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.92 - 3.37i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.39 + 3.92i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.492 + 1.07i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 6.07T + 47T^{2} \) |
| 53 | \( 1 + (-5.16 + 3.32i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.0113 - 0.00729i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (4.91 + 10.7i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.794 + 5.52i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.689 - 4.79i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-7.49 + 2.20i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (11.7 + 7.52i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.23 - 8.34i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.28 + 9.37i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-11.2 + 12.9i)T + (-13.8 - 96.0i)T^{2} \) |
| 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
−9.532581223165003215647221072271, −8.819788103254775744241346296502, −8.199108135900152950054607588124, −7.31931964194159470961556159004, −6.15715067795676353644447812424, −5.76125759367150914676757776514, −4.76962020844056194207125638219, −3.45337560319533520732024755714, −2.19016456114071075384161137524, −0.54914847270626411263400487380,
2.06581946936315364439875566708, 2.58651568457712452023712243098, 4.02752738741795589523146044800, 4.61135693692686944550091185476, 5.88055942655319923374462704347, 6.83733019418251835474776259013, 7.63536172131827962566205605388, 9.032017562907035185617736487075, 9.646875237693320583632720078950, 10.15228846043331987684562091634
