| 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
−9.539994197945387887613938468583, −8.494000666770645176139428578355, −7.909083286986748247693331349605, −7.43650287619915426417529815968, −5.95626387685396917610145178112, −5.24818244057855168494969354672, −4.56206012994050241405905097852, −2.79981091839100945327314407302, −1.21778507648902232933990962371, −0.32008696663499598047873338053,
2.17785422239969813874528408343, 3.00574124831443568286515210420, 4.08754866060410649584591413758, 5.47661718468876651359101602324, 6.35241325759384748407829890446, 7.36220081148406583135054617115, 7.73040616570111273032087804241, 8.989490447758857894052819910207, 10.07485798600522852804275800666, 10.32545332018710735754355251642
