| L(s) = 1 | + (0.672 − 1.30i)2-s + (−0.0891 − 0.125i)4-s + (0.637 − 2.62i)5-s + (0.985 + 0.341i)7-s + (2.68 − 0.385i)8-s + (−2.99 − 2.59i)10-s + (0.0523 − 1.09i)11-s + (0.273 + 0.790i)13-s + (1.10 − 1.05i)14-s + (1.40 − 4.04i)16-s + (1.94 + 4.25i)17-s + (−2.86 − 1.30i)19-s + (−0.385 + 0.154i)20-s + (−1.39 − 0.806i)22-s + (−0.391 − 4.77i)23-s + ⋯ |
| L(s) = 1 | + (0.475 − 0.922i)2-s + (−0.0445 − 0.0626i)4-s + (0.285 − 1.17i)5-s + (0.372 + 0.128i)7-s + (0.948 − 0.136i)8-s + (−0.948 − 0.821i)10-s + (0.0157 − 0.331i)11-s + (0.0758 + 0.219i)13-s + (0.296 − 0.282i)14-s + (0.350 − 1.01i)16-s + (0.470 + 1.03i)17-s + (−0.657 − 0.300i)19-s + (−0.0862 + 0.0345i)20-s + (−0.297 − 0.171i)22-s + (−0.0815 − 0.996i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50775 - 1.70180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50775 - 1.70180i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (0.391 + 4.77i)T \) |
| good | 2 | \( 1 + (-0.672 + 1.30i)T + (-1.16 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-0.637 + 2.62i)T + (-4.44 - 2.29i)T^{2} \) |
| 7 | \( 1 + (-0.985 - 0.341i)T + (5.50 + 4.32i)T^{2} \) |
| 11 | \( 1 + (-0.0523 + 1.09i)T + (-10.9 - 1.04i)T^{2} \) |
| 13 | \( 1 + (-0.273 - 0.790i)T + (-10.2 + 8.03i)T^{2} \) |
| 17 | \( 1 + (-1.94 - 4.25i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (2.86 + 1.30i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.31 + 1.64i)T + (9.48 + 27.4i)T^{2} \) |
| 31 | \( 1 + (0.469 + 0.187i)T + (22.4 + 21.3i)T^{2} \) |
| 37 | \( 1 + (0.606 - 2.06i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (8.46 + 2.05i)T + (36.4 + 18.7i)T^{2} \) |
| 43 | \( 1 + (-4.37 - 10.9i)T + (-31.1 + 29.6i)T^{2} \) |
| 47 | \( 1 + (-10.9 + 6.30i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.36 + 6.19i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (4.38 - 1.51i)T + (46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (-1.61 - 2.05i)T + (-14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (1.38 - 0.0658i)T + (66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (-4.43 - 6.89i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.42 - 3.11i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (2.89 - 15.0i)T + (-73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-3.17 - 13.0i)T + (-73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (2.12 - 14.8i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-5.31 + 5.57i)T + (-4.61 - 96.8i)T^{2} \) |
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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
−10.59607355896872973000982022354, −9.675945164457337960529536007263, −8.575528809747378019585476510836, −8.077727111009855070973612964157, −6.69460239639474088794123735083, −5.48804025213149740179007148290, −4.59738909156879656687989740994, −3.76183034566499277288700320864, −2.33497066049506711992177841430, −1.25491607098575101010991996113,
1.86982463205141308276594105424, 3.26588883194279729496956740314, 4.57875325539837647004362765908, 5.58202757504173298047408614059, 6.34245179174857097686733432964, 7.31555473303609273006324990136, 7.64115906476358443472320510006, 9.068527842739517624070209923559, 10.20177122551189488903336434765, 10.71794387327147891287096108517
