| L(s) = 1 | + (−0.987 − 0.156i)2-s + (1.64 + 0.556i)3-s + (0.951 + 0.309i)4-s + (0.545 + 2.16i)5-s + (−1.53 − 0.806i)6-s + (−1.41 + 1.41i)7-s + (−0.891 − 0.453i)8-s + (2.37 + 1.82i)9-s + (−0.199 − 2.22i)10-s + (−2.37 − 3.26i)11-s + (1.38 + 1.03i)12-s + (0.301 + 1.90i)13-s + (1.62 − 1.17i)14-s + (−0.313 + 3.86i)15-s + (0.809 + 0.587i)16-s + (1.78 − 3.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 − 0.110i)2-s + (0.946 + 0.321i)3-s + (0.475 + 0.154i)4-s + (0.243 + 0.969i)5-s + (−0.625 − 0.329i)6-s + (−0.535 + 0.535i)7-s + (−0.315 − 0.160i)8-s + (0.793 + 0.608i)9-s + (−0.0629 − 0.704i)10-s + (−0.715 − 0.984i)11-s + (0.400 + 0.299i)12-s + (0.0836 + 0.528i)13-s + (0.433 − 0.314i)14-s + (−0.0809 + 0.996i)15-s + (0.202 + 0.146i)16-s + (0.432 − 0.849i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00355 + 0.402066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00355 + 0.402066i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.987 + 0.156i)T \) |
| 3 | \( 1 + (-1.64 - 0.556i)T \) |
| 5 | \( 1 + (-0.545 - 2.16i)T \) |
| good | 7 | \( 1 + (1.41 - 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.37 + 3.26i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.301 - 1.90i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.78 + 3.50i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-6.42 + 2.08i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.317 - 2.00i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.79 + 5.53i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.88 + 8.86i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (6.31 - 1.00i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.756 + 1.04i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (6.68 + 6.68i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.18 - 0.601i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-4.96 - 9.73i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (3.81 + 2.77i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.433 - 0.315i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.35 + 0.688i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-0.520 - 0.169i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.37 - 1.16i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-10.2 - 3.31i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.00 + 1.53i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (11.4 - 8.34i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (4.82 + 9.47i)T + (-57.0 + 78.4i)T^{2} \) |
| 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
−13.53967773507766230144149022655, −11.87930835570100602782141920794, −10.91258743979589331067296716240, −9.800031091777191161019396502451, −9.270897360059107900510658285506, −7.987685510414833916083035552474, −7.08215429309056049082492626150, −5.64079945519213346011849894264, −3.39793774825110711696890613815, −2.51722321298787086042020377390,
1.51171441837320764984601478835, 3.37351413117533021946057162709, 5.18079529151263392511359541632, 6.85815671903443871962661844473, 7.84069512347685081734873423054, 8.672313865358792997622065571127, 9.802868809813203464747647296012, 10.30928743446088650685888438144, 12.28058947488345700226754872731, 12.79539429388820522265096219135
