| L(s) = 1 | + (−1.23 + 3.80i)2-s + (17.4 + 12.6i)3-s + (−12.9 − 9.40i)4-s + (53.8 + 14.9i)5-s + (−69.8 + 50.7i)6-s + 157.·7-s + (51.7 − 37.6i)8-s + (68.8 + 211. i)9-s + (−123. + 186. i)10-s + (−28.9 + 89.1i)11-s + (−106. − 328. i)12-s + (−194. − 599. i)13-s + (−194. + 599. i)14-s + (751. + 944. i)15-s + (79.1 + 243. i)16-s + (−905. + 657. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (1.12 + 0.813i)3-s + (−0.404 − 0.293i)4-s + (0.963 + 0.267i)5-s + (−0.792 + 0.575i)6-s + 1.21·7-s + (0.286 − 0.207i)8-s + (0.283 + 0.872i)9-s + (−0.390 + 0.589i)10-s + (−0.0721 + 0.222i)11-s + (−0.213 − 0.658i)12-s + (−0.319 − 0.984i)13-s + (−0.265 + 0.817i)14-s + (0.861 + 1.08i)15-s + (0.0772 + 0.237i)16-s + (−0.759 + 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0244 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0244 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.71970 + 1.76232i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71970 + 1.76232i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 5 | \( 1 + (-53.8 - 14.9i)T \) |
| good | 3 | \( 1 + (-17.4 - 12.6i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 157.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (28.9 - 89.1i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (194. + 599. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (905. - 657. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.50e3 - 1.09e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (144. - 444. i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (4.99e3 + 3.62e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-5.63e3 + 4.09e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (5.04e3 + 1.55e4i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-2.64e3 - 8.12e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 3.85e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.21e4 - 8.81e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.45e4 - 1.05e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.01e4 + 3.12e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-4.83e3 + 1.48e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.13e4 - 2.27e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.31e4 - 1.68e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (2.58e4 - 7.96e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.16e4 + 5.20e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-5.34e4 + 3.88e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (3.39e3 - 1.04e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (3.95e4 + 2.87e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−14.80710476032388916670325051540, −14.22012984420971658693487256424, −13.02511142884778985250935341547, −10.80227239663854607179292677079, −9.840650169865250460188784538880, −8.720100718174069440314018120545, −7.74710716380796475929994468429, −5.80554788741633504348654197704, −4.29159678406969184283349024402, −2.19486332509522075039876803669,
1.54613695583009522856710982887, 2.47067989920979018563195017488, 4.76203654222052817440017369125, 6.98040267484624555269123083096, 8.489476162045188381368988043645, 9.090790192865585516379034805704, 10.70996976502156245673308011686, 12.03913195444825118543968784807, 13.37864851975453276437540169757, 13.88484303636173309258997417286
