| L(s) = 1 | + (−0.805 + 2.40i)2-s + (−0.153 − 0.0861i)3-s + (−3.54 − 2.67i)4-s + (−0.680 + 1.02i)5-s + (0.330 − 0.299i)6-s + (1.03 − 2.28i)7-s + (5.09 − 3.49i)8-s + (−1.54 − 2.53i)9-s + (−1.92 − 2.46i)10-s + (−2.29 − 0.147i)11-s + (0.313 + 0.715i)12-s + (3.73 + 4.31i)13-s + (4.65 + 4.32i)14-s + (0.192 − 0.0987i)15-s + (1.87 + 6.54i)16-s + (0.975 − 0.430i)17-s + ⋯ |
| L(s) = 1 | + (−0.569 + 1.70i)2-s + (−0.0885 − 0.0497i)3-s + (−1.77 − 1.33i)4-s + (−0.304 + 0.458i)5-s + (0.135 − 0.122i)6-s + (0.391 − 0.863i)7-s + (1.80 − 1.23i)8-s + (−0.514 − 0.845i)9-s + (−0.607 − 0.779i)10-s + (−0.693 − 0.0444i)11-s + (0.0903 + 0.206i)12-s + (1.03 + 1.19i)13-s + (1.24 + 1.15i)14-s + (0.0497 − 0.0255i)15-s + (0.467 + 1.63i)16-s + (0.236 − 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.170566 + 0.791849i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170566 + 0.791849i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 + (-15.7 - 27.0i)T \) |
| good | 2 | \( 1 + (0.805 - 2.40i)T + (-1.59 - 1.20i)T^{2} \) |
| 3 | \( 1 + (0.153 + 0.0861i)T + (1.56 + 2.56i)T^{2} \) |
| 5 | \( 1 + (0.680 - 1.02i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (-1.03 + 2.28i)T + (-4.61 - 5.26i)T^{2} \) |
| 11 | \( 1 + (2.29 + 0.147i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (-3.73 - 4.31i)T + (-1.86 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.975 + 0.430i)T + (11.4 - 12.5i)T^{2} \) |
| 19 | \( 1 + (-0.853 - 0.652i)T + (4.98 + 18.3i)T^{2} \) |
| 23 | \( 1 + (0.310 + 2.92i)T + (-22.4 + 4.82i)T^{2} \) |
| 29 | \( 1 + (2.97 - 4.30i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (-4.90 - 7.30i)T + (-11.7 + 28.7i)T^{2} \) |
| 37 | \( 1 + (-1.22 + 0.303i)T + (32.7 - 17.2i)T^{2} \) |
| 41 | \( 1 + (8.48 + 0.762i)T + (40.3 + 7.30i)T^{2} \) |
| 43 | \( 1 + (-3.51 - 4.82i)T + (-13.1 + 40.9i)T^{2} \) |
| 47 | \( 1 + (-1.40 - 4.68i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-1.18 + 1.08i)T + (4.57 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-8.33 - 0.641i)T + (58.3 + 9.02i)T^{2} \) |
| 61 | \( 1 + (-1.43 - 1.22i)T + (9.90 + 60.1i)T^{2} \) |
| 67 | \( 1 + (-3.86 + 3.32i)T + (10.0 - 66.2i)T^{2} \) |
| 71 | \( 1 + (3.48 - 2.84i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (3.87 + 2.76i)T + (23.6 + 69.0i)T^{2} \) |
| 79 | \( 1 + (-9.82 + 1.45i)T + (75.6 - 22.9i)T^{2} \) |
| 83 | \( 1 + (2.89 - 0.898i)T + (68.4 - 46.9i)T^{2} \) |
| 89 | \( 1 + (2.23 + 1.46i)T + (35.1 + 81.7i)T^{2} \) |
| 97 | \( 1 + (-2.47 + 0.126i)T + (96.4 - 9.91i)T^{2} \) |
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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
−10.16541340910437683774813942536, −9.116353407926264915270726703087, −8.567387225408221882524206994429, −7.71714034074771098854512785862, −6.92613331026135332422894836650, −6.46281297064172369942425721068, −5.46575203723212707818724853321, −4.47870626924605759569453517857, −3.39407473027739473009204536156, −1.07805204899359639181296886453,
0.58051289949415841007762679631, 2.06496987271433280160770617829, 2.86314779000101078458027223390, 3.95720602964027211036859360934, 5.09786476315039813944996638777, 5.78041231832392999846653390490, 7.81662387619998332471329736018, 8.286544702965635346293106302695, 8.796141082595147915595232979301, 9.936961008520632119493773250120
