| 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 | \( 1 + (4.82 - 9.47i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−12.79539429388820522265096219135, −12.28058947488345700226754872731, −10.30928743446088650685888438144, −9.802868809813203464747647296012, −8.672313865358792997622065571127, −7.84069512347685081734873423054, −6.85815671903443871962661844473, −5.18079529151263392511359541632, −3.37351413117533021946057162709, −1.51171441837320764984601478835,
2.51722321298787086042020377390, 3.39793774825110711696890613815, 5.64079945519213346011849894264, 7.08215429309056049082492626150, 7.987685510414833916083035552474, 9.270897360059107900510658285506, 9.800031091777191161019396502451, 10.91258743979589331067296716240, 11.87930835570100602782141920794, 13.53967773507766230144149022655
