| L(s) = 1 | + (−0.349 − 1.37i)2-s + (1.71 − 0.207i)3-s + (−1.75 + 0.958i)4-s + (−2.21 − 2.52i)5-s + (−0.885 − 2.28i)6-s + (4.47 + 0.588i)7-s + (1.92 + 2.06i)8-s + (2.91 − 0.712i)9-s + (−2.68 + 3.91i)10-s + (−1.40 − 2.85i)11-s + (−2.81 + 2.01i)12-s + (0.366 + 1.07i)13-s + (−0.757 − 6.33i)14-s + (−4.32 − 3.88i)15-s + (2.16 − 3.36i)16-s + (0.812 + 0.812i)17-s + ⋯ |
| L(s) = 1 | + (−0.247 − 0.968i)2-s + (0.992 − 0.119i)3-s + (−0.877 + 0.479i)4-s + (−0.989 − 1.12i)5-s + (−0.361 − 0.932i)6-s + (1.68 + 0.222i)7-s + (0.681 + 0.731i)8-s + (0.971 − 0.237i)9-s + (−0.848 + 1.23i)10-s + (−0.424 − 0.860i)11-s + (−0.813 + 0.581i)12-s + (0.101 + 0.299i)13-s + (−0.202 − 1.69i)14-s + (−1.11 − 1.00i)15-s + (0.540 − 0.841i)16-s + (0.197 + 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.736182 - 1.44274i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.736182 - 1.44274i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.349 + 1.37i)T \) |
| 3 | \( 1 + (-1.71 + 0.207i)T \) |
| good | 5 | \( 1 + (2.21 + 2.52i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-4.47 - 0.588i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.40 + 2.85i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-0.366 - 1.07i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-0.812 - 0.812i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.674 + 3.39i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.460 + 3.49i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (7.12 - 0.466i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (4.47 - 2.58i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.859 + 4.31i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.07 - 8.20i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-6.38 + 3.15i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (3.45 - 0.925i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.03 + 3.04i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-9.14 - 10.4i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.0872 + 1.33i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (12.5 + 6.19i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (2.10 + 5.08i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.01 + 4.86i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-3.56 - 13.3i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-12.2 - 10.7i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-9.22 + 3.82i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (1.92 + 1.10i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.63871789804872699718529333176, −9.096419718684618802786187419972, −8.806603767565681827065950964427, −7.998699636109778621502538141249, −7.58508648918618314193657081690, −5.24112315251125647966804502191, −4.48214113990565730142539069581, −3.63035983281931376920985988680, −2.19655606447183263754297114916, −1.01332547965302537808983896757,
1.86249883184878747231558443335, 3.58933042049296331767383447981, 4.39210084984785294612494582892, 5.47422180731304885908130952763, 7.11812403409486501115677876609, 7.68673803369333996026890268147, 7.891913713078520830775050552246, 9.013348587935193914618187967657, 10.10700754301861662046335490899, 10.77561866591220473846835582213
