| L(s) = 1 | + (−1.09 − 1.34i)3-s + (1.20 + 1.59i)4-s + (−0.929 + 0.790i)7-s + (−0.600 + 2.93i)9-s + (0.829 − 3.36i)12-s + (2.59 + 2.5i)13-s + (−1.11 + 3.84i)16-s + (−0.769 + 2.87i)19-s + (2.07 + 0.381i)21-s + (−2.32 + 4.42i)25-s + (4.60 − 2.41i)27-s + (−2.38 − 0.536i)28-s + (10.6 + 3.31i)31-s + (−5.42 + 2.57i)36-s + (−3.39 − 3.68i)37-s + ⋯ |
| L(s) = 1 | + (−0.632 − 0.774i)3-s + (0.600 + 0.799i)4-s + (−0.351 + 0.298i)7-s + (−0.200 + 0.979i)9-s + (0.239 − 0.970i)12-s + (0.720 + 0.693i)13-s + (−0.278 + 0.960i)16-s + (−0.176 + 0.659i)19-s + (0.453 + 0.0831i)21-s + (−0.464 + 0.885i)25-s + (0.885 − 0.464i)27-s + (−0.450 − 0.101i)28-s + (1.90 + 0.594i)31-s + (−0.903 + 0.428i)36-s + (−0.558 − 0.605i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05097 + 0.548313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05097 + 0.548313i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.09 + 1.34i)T \) |
| 13 | \( 1 + (-2.59 - 2.5i)T \) |
| good | 2 | \( 1 + (-1.20 - 1.59i)T^{2} \) |
| 5 | \( 1 + (2.32 - 4.42i)T^{2} \) |
| 7 | \( 1 + (0.929 - 0.790i)T + (1.12 - 6.90i)T^{2} \) |
| 11 | \( 1 + (-4.31 - 10.1i)T^{2} \) |
| 17 | \( 1 + (16.7 + 2.72i)T^{2} \) |
| 19 | \( 1 + (0.769 - 2.87i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-23.1 + 17.4i)T^{2} \) |
| 31 | \( 1 + (-10.6 - 3.31i)T + (25.5 + 17.6i)T^{2} \) |
| 37 | \( 1 + (3.39 + 3.68i)T + (-2.97 + 36.8i)T^{2} \) |
| 41 | \( 1 + (40.1 + 8.20i)T^{2} \) |
| 43 | \( 1 + (-9.70 + 0.391i)T + (42.8 - 3.46i)T^{2} \) |
| 47 | \( 1 + (11.2 - 45.6i)T^{2} \) |
| 53 | \( 1 + (18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-31.5 - 49.8i)T^{2} \) |
| 61 | \( 1 + (5.08 - 10.7i)T + (-38.5 - 47.2i)T^{2} \) |
| 67 | \( 1 + (2.08 + 14.7i)T + (-64.3 + 18.6i)T^{2} \) |
| 71 | \( 1 + (-22.4 - 67.3i)T^{2} \) |
| 73 | \( 1 + (0.441 - 7.29i)T + (-72.4 - 8.79i)T^{2} \) |
| 79 | \( 1 + (-1.98 + 16.3i)T + (-76.7 - 18.9i)T^{2} \) |
| 83 | \( 1 + (55.0 + 62.1i)T^{2} \) |
| 89 | \( 1 + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.200 + 9.93i)T + (-96.9 - 3.90i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.22761276353891938520284932795, −10.47274303158241194943796163839, −9.089373206768177639089745430684, −8.166770173158095142081916492640, −7.34265142658943708694915539082, −6.46970227065509184353805560648, −5.80363230458295797139580318739, −4.27973495724535219226436116077, −2.94811620547460675378247198585, −1.66448844483583352951555883528,
0.789359332911342529774983451004, 2.77537644710374544058665364100, 4.12849714711792065118661603696, 5.21000123992461685880425752730, 6.14861029419038554686751266978, 6.72279893855188187824554999872, 8.125883325716443419892079482274, 9.324875529165645721117949753010, 10.13627450152175583850453240498, 10.64203317742566721148632166090
