L(s) = 1 | + (−0.818 − 0.422i)2-s + (−0.786 − 0.618i)3-s + (−0.667 − 0.937i)4-s + (2.57 + 2.45i)5-s + (0.382 + 0.838i)6-s + (−2.33 − 1.25i)7-s + (0.413 + 2.87i)8-s + (0.235 + 0.971i)9-s + (−1.07 − 3.09i)10-s + (1.66 − 0.859i)11-s + (−0.0547 + 1.14i)12-s + (−4.13 + 4.77i)13-s + (1.38 + 2.00i)14-s + (−0.506 − 3.52i)15-s + (0.121 − 0.352i)16-s + (−6.90 − 0.659i)17-s + ⋯ |
L(s) = 1 | + (−0.579 − 0.298i)2-s + (−0.453 − 0.356i)3-s + (−0.333 − 0.468i)4-s + (1.15 + 1.09i)5-s + (0.156 + 0.342i)6-s + (−0.881 − 0.472i)7-s + (0.146 + 1.01i)8-s + (0.0785 + 0.323i)9-s + (−0.339 − 0.979i)10-s + (0.502 − 0.259i)11-s + (−0.0158 + 0.331i)12-s + (−1.14 + 1.32i)13-s + (0.369 + 0.536i)14-s + (−0.130 − 0.909i)15-s + (0.0304 − 0.0880i)16-s + (−1.67 − 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.452345 + 0.334369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.452345 + 0.334369i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (2.33 + 1.25i)T \) |
| 23 | \( 1 + (0.621 - 4.75i)T \) |
good | 2 | \( 1 + (0.818 + 0.422i)T + (1.16 + 1.62i)T^{2} \) |
| 5 | \( 1 + (-2.57 - 2.45i)T + (0.237 + 4.99i)T^{2} \) |
| 11 | \( 1 + (-1.66 + 0.859i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (4.13 - 4.77i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.90 + 0.659i)T + (16.6 + 3.21i)T^{2} \) |
| 19 | \( 1 + (-3.32 + 0.317i)T + (18.6 - 3.59i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 5.39i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.474 - 0.190i)T + (22.4 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-0.772 - 3.18i)T + (-32.8 + 16.9i)T^{2} \) |
| 41 | \( 1 + (-1.00 + 0.294i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.52 - 10.6i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (1.36 + 2.35i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.77 + 1.11i)T + (49.2 - 19.6i)T^{2} \) |
| 59 | \( 1 + (-0.0403 - 0.116i)T + (-46.3 + 36.4i)T^{2} \) |
| 61 | \( 1 + (-1.78 + 1.40i)T + (14.3 - 59.2i)T^{2} \) |
| 67 | \( 1 + (-0.498 - 10.4i)T + (-66.6 + 6.36i)T^{2} \) |
| 71 | \( 1 + (-4.06 + 2.61i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.95 + 6.96i)T + (-23.8 + 68.9i)T^{2} \) |
| 79 | \( 1 + (14.0 + 2.71i)T + (73.3 + 29.3i)T^{2} \) |
| 83 | \( 1 + (3.15 + 0.925i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (16.1 - 6.46i)T + (64.4 - 61.4i)T^{2} \) |
| 97 | \( 1 + (10.8 - 3.18i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.09734965499944027998215459149, −10.03448660291354663792566224586, −9.681920110907842446529167099368, −8.897660099595785073108563846115, −7.12538372182732181405549932997, −6.68805904434945468987342147336, −5.78452394667476714361429054289, −4.55455354930013057514972169238, −2.76499799637669932342895626236, −1.63201328745296057182461016082,
0.43327520919790803640312860124, 2.54098654924484563755248875195, 4.20550768045499449988579374566, 5.17795399677668318791158225285, 6.12079713286549550397291351236, 7.09118951723192918585076491840, 8.453865703687948842101851578476, 9.083796642954207990839201995629, 9.788978439327944442041556768604, 10.26333758441204303104790301499