L(s) = 1 | + (−0.815 + 2.43i)2-s + (1.99 + 1.11i)3-s + (−3.66 − 2.76i)4-s + (0.269 − 0.407i)5-s + (−4.34 + 3.93i)6-s + (0.302 − 0.667i)7-s + (5.48 − 3.76i)8-s + (1.15 + 1.89i)9-s + (0.771 + 0.988i)10-s + (4.53 + 0.290i)11-s + (−4.20 − 9.61i)12-s + (0.900 + 1.04i)13-s + (1.37 + 1.27i)14-s + (0.993 − 0.509i)15-s + (2.16 + 7.58i)16-s + (−3.64 + 1.60i)17-s + ⋯ |
L(s) = 1 | + (−0.576 + 1.72i)2-s + (1.15 + 0.646i)3-s + (−1.83 − 1.38i)4-s + (0.120 − 0.182i)5-s + (−1.77 + 1.60i)6-s + (0.114 − 0.252i)7-s + (1.93 − 1.32i)8-s + (0.385 + 0.633i)9-s + (0.243 + 0.312i)10-s + (1.36 + 0.0876i)11-s + (−1.21 − 2.77i)12-s + (0.249 + 0.288i)13-s + (0.368 + 0.341i)14-s + (0.256 − 0.131i)15-s + (0.541 + 1.89i)16-s + (−0.883 + 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.256711 + 1.68236i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.256711 + 1.68236i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 983 | \( 1 + (-18.9 - 24.9i)T \) |
good | 2 | \( 1 + (0.815 - 2.43i)T + (-1.59 - 1.20i)T^{2} \) |
| 3 | \( 1 + (-1.99 - 1.11i)T + (1.56 + 2.56i)T^{2} \) |
| 5 | \( 1 + (-0.269 + 0.407i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (-0.302 + 0.667i)T + (-4.61 - 5.26i)T^{2} \) |
| 11 | \( 1 + (-4.53 - 0.290i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (-0.900 - 1.04i)T + (-1.86 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.64 - 1.60i)T + (11.4 - 12.5i)T^{2} \) |
| 19 | \( 1 + (-3.65 - 2.79i)T + (4.98 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.465 - 4.39i)T + (-22.4 + 4.82i)T^{2} \) |
| 29 | \( 1 + (0.311 - 0.450i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (-0.0749 - 0.111i)T + (-11.7 + 28.7i)T^{2} \) |
| 37 | \( 1 + (4.98 - 1.23i)T + (32.7 - 17.2i)T^{2} \) |
| 41 | \( 1 + (-1.33 - 0.120i)T + (40.3 + 7.30i)T^{2} \) |
| 43 | \( 1 + (-2.35 - 3.23i)T + (-13.1 + 40.9i)T^{2} \) |
| 47 | \( 1 + (2.50 + 8.37i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-2.40 + 2.20i)T + (4.57 - 52.8i)T^{2} \) |
| 59 | \( 1 + (13.7 + 1.05i)T + (58.3 + 9.02i)T^{2} \) |
| 61 | \( 1 + (0.904 + 0.767i)T + (9.90 + 60.1i)T^{2} \) |
| 67 | \( 1 + (-7.74 + 6.66i)T + (10.0 - 66.2i)T^{2} \) |
| 71 | \( 1 + (5.15 - 4.20i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-9.34 - 6.68i)T + (23.6 + 69.0i)T^{2} \) |
| 79 | \( 1 + (-3.78 + 0.560i)T + (75.6 - 22.9i)T^{2} \) |
| 83 | \( 1 + (-0.781 + 0.242i)T + (68.4 - 46.9i)T^{2} \) |
| 89 | \( 1 + (-5.71 - 3.76i)T + (35.1 + 81.7i)T^{2} \) |
| 97 | \( 1 + (5.66 - 0.290i)T + (96.4 - 9.91i)T^{2} \) |
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
−9.600939562879895889247747361555, −9.345326266991517893969676377323, −8.703074478547614136529344054807, −7.979195265587244494065745855829, −7.11304382961621220910783121408, −6.36227583604904710580858988973, −5.32916088253938846132424306320, −4.28602553769022889788128343727, −3.52705417764941343620100714647, −1.47565834719340534481561285813,
0.968297217043251410731599743750, 2.09797016874898160003554125611, 2.80389524768495728970678328318, 3.67878330256501913372941715660, 4.71571641423138959071113455020, 6.45880055053483172653873158454, 7.45968002014112104544855511771, 8.455383155220300371927617128684, 8.977384145852221444776199347295, 9.409795062426625454262309445924