| L(s) = 1 | + (0.934 − 1.06i)2-s + (1.41 − 0.714i)3-s + (−0.252 − 1.98i)4-s + (1.27 + 2.87i)5-s + (0.568 − 2.17i)6-s + (2.43 + 1.45i)7-s + (−2.34 − 1.58i)8-s + (−0.284 + 0.383i)9-s + (4.24 + 1.33i)10-s + (−4.13 − 3.22i)11-s + (−1.77 − 2.63i)12-s + (3.96 − 3.77i)13-s + (3.82 − 1.21i)14-s + (3.86 + 3.17i)15-s + (−3.87 + 1.00i)16-s + (2.50 + 3.05i)17-s + ⋯ |
| L(s) = 1 | + (0.660 − 0.750i)2-s + (0.819 − 0.412i)3-s + (−0.126 − 0.992i)4-s + (0.570 + 1.28i)5-s + (0.232 − 0.887i)6-s + (0.919 + 0.551i)7-s + (−0.827 − 0.560i)8-s + (−0.0947 + 0.127i)9-s + (1.34 + 0.422i)10-s + (−1.24 − 0.972i)11-s + (−0.512 − 0.760i)12-s + (1.09 − 1.04i)13-s + (1.02 − 0.325i)14-s + (0.998 + 0.819i)15-s + (−0.968 + 0.250i)16-s + (0.607 + 0.740i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.45714 - 1.36412i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.45714 - 1.36412i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.934 + 1.06i)T \) |
| good | 3 | \( 1 + (-1.41 + 0.714i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-1.27 - 2.87i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-2.43 - 1.45i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (4.13 + 3.22i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.96 + 3.77i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-2.50 - 3.05i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.759 + 4.37i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-2.78 - 1.31i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-5.27 - 2.98i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (9.20 - 1.83i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (7.59 + 4.81i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-2.98 - 3.29i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (1.04 + 3.17i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (10.5 + 5.64i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (4.35 - 2.46i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (5.47 - 5.75i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.721 - 9.77i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-5.29 - 4.56i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-5.22 - 0.775i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (13.7 - 8.23i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-0.955 - 0.289i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-5.58 + 3.53i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-6.30 + 2.98i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (10.1 + 6.76i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.80511312843453764027025732801, −10.37677044791999355197603279134, −8.834279154087535134565780459770, −8.260557758318767042108550247218, −7.07760939635425464721497683247, −5.75979958627205019443862113669, −5.30198301622391211918835769749, −3.27649895185659900467093871009, −2.87707514994234698411261190029, −1.75125171201913947925034615660,
1.86190702267067042665493027682, 3.48709303714390907328427268549, 4.62514886829626487941457050032, 5.07818414269560691569451772630, 6.31437726925305188178413704667, 7.68659329157208401881973591632, 8.248077329694749800360550165119, 9.055354643108393799030505667849, 9.802909327840960035725874657714, 11.13655174765675518449548132824
