| L(s) = 1 | + (0.309 − 0.951i)2-s + (0.978 + 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (−3.12 + 1.39i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (−0.562 + 5.35i)11-s + (−0.669 − 0.743i)12-s + (−1.32 + 1.46i)13-s + (0.357 + 3.40i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.0240 − 0.228i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.564 + 0.120i)3-s + (−0.404 − 0.293i)4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s + (−1.18 + 0.526i)7-s + (−0.286 + 0.207i)8-s + (0.304 + 0.135i)9-s + (−0.309 + 0.0657i)10-s + (−0.169 + 1.61i)11-s + (−0.193 − 0.214i)12-s + (−0.366 + 0.406i)13-s + (0.0956 + 0.909i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.00582 − 0.0554i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.886593 + 0.603401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.886593 + 0.603401i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (2.03 - 5.18i)T \) |
| good | 7 | \( 1 + (3.12 - 1.39i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.562 - 5.35i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (1.32 - 1.46i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.0240 + 0.228i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.23 - 1.37i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.61 - 1.90i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.55 - 4.77i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-5.67 + 9.82i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.80 + 0.595i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-6.06 - 6.73i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (2.26 + 6.97i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.92 + 2.63i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (7.11 + 1.51i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 9.65T + 61T^{2} \) |
| 67 | \( 1 + (-1.83 - 3.17i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.40 - 1.95i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (1.02 - 9.74i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.788 - 7.49i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (15.4 - 3.28i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (1.04 + 0.762i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.95 - 3.60i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.926801039213859563547580201063, −9.539711540469983404114226834636, −8.931956441480833324646276459444, −7.71826734522021277002928279353, −6.93485219387309011822063578911, −5.72227843755685297464795934106, −4.69682816310486894937539847359, −3.82997992406518534475884957433, −2.79425520302382218864546045331, −1.78556434711117493917738711871,
0.42235791765237351597301335646, 2.82456162338451848060372592089, 3.43167981303806404565136486610, 4.45214970115537800237353484358, 5.98097059226468837718835512530, 6.32644335432320828231350307669, 7.52230442174984189826233388964, 7.960205941836630796915550227202, 9.015689993893734619145752092644, 9.740562060252235262400011427109
