| L(s) = 1 | + (0.766 + 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (0.939 + 0.342i)6-s + (−0.965 − 1.67i)7-s + (−0.500 + 0.866i)8-s + (0.766 − 0.642i)9-s + (0.766 − 0.642i)10-s + (0.732 − 1.26i)11-s + (0.499 + 0.866i)12-s + (4.09 + 1.49i)13-s + (0.335 − 1.90i)14-s + (−0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (5.27 + 4.42i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.542 − 0.197i)3-s + (0.0868 + 0.492i)4-s + (0.0776 − 0.440i)5-s + (0.383 + 0.139i)6-s + (−0.364 − 0.631i)7-s + (−0.176 + 0.306i)8-s + (0.255 − 0.214i)9-s + (0.242 − 0.203i)10-s + (0.220 − 0.382i)11-s + (0.144 + 0.250i)12-s + (1.13 + 0.413i)13-s + (0.0895 − 0.508i)14-s + (−0.0448 − 0.254i)15-s + (−0.234 + 0.0855i)16-s + (1.27 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.40490 + 0.0939238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.40490 + 0.0939238i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (-3.99 + 1.74i)T \) |
| good | 7 | \( 1 + (0.965 + 1.67i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.732 + 1.26i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.09 - 1.49i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.27 - 4.42i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.26 + 7.18i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.03 - 2.54i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.44 - 2.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + (7.54 - 2.74i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0773 - 0.438i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (6.66 - 5.59i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (1.32 + 7.49i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (5.40 + 4.53i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.10 - 11.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.817 + 0.685i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.09 - 11.8i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.10 + 1.85i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.20 + 0.436i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.57 - 9.65i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (11.4 + 4.17i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-9.63 - 8.08i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.71713818106832756132624999808, −9.816507165103146396404484451302, −8.654092274277978314165579539512, −8.205378764664678510337557876006, −7.02298886140374451644775961441, −6.29428628960892725411381346695, −5.20669392991894193131387267177, −3.92513633424817634973726562347, −3.26833641790919182847487070003, −1.39129782200666623570043280689,
1.67515233031208779750503441305, 3.15674577581013138982149479273, 3.59214732299689472385574126878, 5.20127512855248583906512260088, 5.90045080164048450072383553577, 7.14192633212036261783493811976, 8.059508403480194248208757548809, 9.330020798026958399394831601953, 9.771501308718168096980083534033, 10.72012266752755449466185957826
