| L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.578 + 4.02i)5-s + (0.841 − 0.540i)6-s + (0.415 + 0.909i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (1.68 − 3.69i)10-s + (−4.31 + 1.26i)11-s + (−0.959 + 0.281i)12-s + (−1.31 + 2.88i)13-s + (−0.142 − 0.989i)14-s + (−2.66 − 3.07i)15-s + (0.415 + 0.909i)16-s + (−0.558 + 0.359i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (−0.258 + 1.79i)5-s + (0.343 − 0.220i)6-s + (0.157 + 0.343i)7-s + (−0.231 − 0.267i)8-s + (−0.0474 − 0.329i)9-s + (0.533 − 1.16i)10-s + (−1.30 + 0.382i)11-s + (−0.276 + 0.0813i)12-s + (−0.365 + 0.800i)13-s + (−0.0380 − 0.264i)14-s + (−0.687 − 0.792i)15-s + (0.103 + 0.227i)16-s + (−0.135 + 0.0870i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.103893 - 0.474412i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103893 - 0.474412i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (4.57 + 1.44i)T \) |
| good | 5 | \( 1 + (0.578 - 4.02i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (4.31 - 1.26i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.31 - 2.88i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.558 - 0.359i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-6.29 - 4.04i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 1.01i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.24 - 4.90i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.681 + 4.74i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.859 + 5.97i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.73 + 6.62i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-2.29 - 5.02i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (3.64 - 7.99i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (7.57 + 8.74i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-5.92 - 1.73i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (1.20 + 0.353i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (11.6 + 7.47i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (3.86 - 8.45i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.37 + 9.54i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (2.66 - 3.07i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.803 - 5.58i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.32545332018710735754355251642, −10.07485798600522852804275800666, −8.989490447758857894052819910207, −7.73040616570111273032087804241, −7.36220081148406583135054617115, −6.35241325759384748407829890446, −5.47661718468876651359101602324, −4.08754866060410649584591413758, −3.00574124831443568286515210420, −2.17785422239969813874528408343,
0.32008696663499598047873338053, 1.21778507648902232933990962371, 2.79981091839100945327314407302, 4.56206012994050241405905097852, 5.24818244057855168494969354672, 5.95626387685396917610145178112, 7.43650287619915426417529815968, 7.909083286986748247693331349605, 8.494000666770645176139428578355, 9.539994197945387887613938468583
