L(s) = 1 | + (−0.309 + 0.951i)2-s + (−1.59 + 1.15i)3-s + (−0.809 − 0.587i)4-s + (−1.29 − 3.97i)5-s + (−0.607 − 1.87i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.267 − 0.824i)9-s + 4.18·10-s + (−3.31 − 0.0152i)11-s + 1.96·12-s + (1.36 − 4.18i)13-s + (0.809 − 0.587i)14-s + (6.65 + 4.83i)15-s + (0.309 + 0.951i)16-s + (−0.340 − 1.04i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.918 + 0.667i)3-s + (−0.404 − 0.293i)4-s + (−0.577 − 1.77i)5-s + (−0.248 − 0.763i)6-s + (−0.305 − 0.222i)7-s + (0.286 − 0.207i)8-s + (0.0892 − 0.274i)9-s + 1.32·10-s + (−0.999 − 0.00460i)11-s + 0.567·12-s + (0.377 − 1.16i)13-s + (0.216 − 0.157i)14-s + (1.71 + 1.24i)15-s + (0.0772 + 0.237i)16-s + (−0.0826 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0411 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0411 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.240616 - 0.230911i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.240616 - 0.230911i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (3.31 + 0.0152i)T \) |
good | 3 | \( 1 + (1.59 - 1.15i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (1.29 + 3.97i)T + (-4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (-1.36 + 4.18i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.340 + 1.04i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.16 - 0.844i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 + (5.91 + 4.29i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.875 - 2.69i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.73 - 4.16i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.916 - 0.666i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 + (-5.81 + 4.22i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.09 + 9.51i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (5.99 + 4.35i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.40 - 10.4i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + (-2.61 - 8.05i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (11.0 + 7.99i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.24 + 6.90i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.99 + 6.12i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 7.52T + 89T^{2} \) |
| 97 | \( 1 + (-0.434 + 1.33i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.86756915206510247085035155680, −11.71706661489702939899186779736, −10.58022104095069144957235746238, −9.641980675891608281538257638494, −8.426973004270237190824061362200, −7.70630829656164125837518106005, −5.77523366909982989092103493895, −5.17203543773667181764765487560, −4.13362120239232191092331892054, −0.37683172127169970493044285486,
2.43100247566667216656776376366, 3.85190251637407995071700714916, 5.88463115902114304932964297463, 6.86168629195247099894011501021, 7.71909999547591903624544570777, 9.359244830160556749594328519086, 10.78025285307222046560525509520, 11.05287327438326442208945661831, 11.97763059582515292508917910552, 12.89852444298815953575434910670