| L(s) = 1 | + (0.856 − 1.83i)2-s + (0.164 + 1.72i)3-s + (−1.35 − 1.61i)4-s + (2.17 − 0.537i)5-s + (3.30 + 1.17i)6-s + (−0.407 + 1.52i)7-s + (−0.215 + 0.0576i)8-s + (−2.94 + 0.566i)9-s + (0.871 − 4.44i)10-s + (1.91 + 1.10i)11-s + (2.56 − 2.60i)12-s + (0.282 + 0.198i)13-s + (2.44 + 2.05i)14-s + (1.28 + 3.65i)15-s + (0.654 − 3.71i)16-s + (2.03 − 4.37i)17-s + ⋯ |
| L(s) = 1 | + (0.605 − 1.29i)2-s + (0.0949 + 0.995i)3-s + (−0.678 − 0.808i)4-s + (0.970 − 0.240i)5-s + (1.35 + 0.479i)6-s + (−0.153 + 0.574i)7-s + (−0.0760 + 0.0203i)8-s + (−0.981 + 0.188i)9-s + (0.275 − 1.40i)10-s + (0.577 + 0.333i)11-s + (0.740 − 0.751i)12-s + (0.0784 + 0.0549i)13-s + (0.653 + 0.548i)14-s + (0.331 + 0.943i)15-s + (0.163 − 0.927i)16-s + (0.494 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.86635 - 0.749789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86635 - 0.749789i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.164 - 1.72i)T \) |
| 5 | \( 1 + (-2.17 + 0.537i)T \) |
| 19 | \( 1 + (3.65 - 2.36i)T \) |
| good | 2 | \( 1 + (-0.856 + 1.83i)T + (-1.28 - 1.53i)T^{2} \) |
| 7 | \( 1 + (0.407 - 1.52i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.91 - 1.10i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.282 - 0.198i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-2.03 + 4.37i)T + (-10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (7.29 + 0.638i)T + (22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (-2.02 - 0.736i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.595 + 1.03i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.43 - 2.43i)T + 37iT^{2} \) |
| 41 | \( 1 + (3.12 + 0.550i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.78 - 0.680i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (11.0 - 5.16i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (4.58 + 0.401i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (0.0709 - 0.0258i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (2.38 - 2.00i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.48 - 11.7i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-8.29 + 9.88i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (3.31 + 4.73i)T + (-24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (-15.0 - 2.66i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.50 + 5.60i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.03 - 5.86i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (12.0 + 5.61i)T + (62.3 + 74.3i)T^{2} \) |
| 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
−11.80706699617006900335292425878, −10.76662388973974582247694121086, −9.831236537623955389129794769379, −9.532248254973154528947827427793, −8.277980559233690027376682654655, −6.31142344888129711924341956572, −5.22658961174914258437805894269, −4.31933875451374003449770734147, −3.09228502060586102281817735613, −1.95930419202307992954755638933,
1.81022897791449273408597076398, 3.71736196658978205358241982744, 5.30245163748031910972856559735, 6.44279526358336459298384801645, 6.51857732365966426277356081615, 7.84397800040891612560539062125, 8.596926305884247504681415196175, 10.00337932537442698319988169678, 11.05291408658582463685389119009, 12.40478093161165119589924915456
