| L(s) = 1 | + (−1.14 − 0.832i)2-s + (1.73 − 0.0244i)3-s + (0.613 + 1.90i)4-s + (0.0414 + 0.711i)5-s + (−2.00 − 1.41i)6-s + (0.543 − 2.29i)7-s + (0.883 − 2.68i)8-s + (2.99 − 0.0848i)9-s + (0.545 − 0.847i)10-s + (0.518 − 0.788i)11-s + (1.10 + 3.28i)12-s + (0.389 − 3.33i)13-s + (−2.53 + 2.17i)14-s + (0.0892 + 1.23i)15-s + (−3.24 + 2.33i)16-s + (0.293 + 0.349i)17-s + ⋯ |
| L(s) = 1 | + (−0.808 − 0.588i)2-s + (0.999 − 0.0141i)3-s + (0.306 + 0.951i)4-s + (0.0185 + 0.318i)5-s + (−0.816 − 0.577i)6-s + (0.205 − 0.867i)7-s + (0.312 − 0.949i)8-s + (0.999 − 0.0282i)9-s + (0.172 − 0.268i)10-s + (0.156 − 0.237i)11-s + (0.320 + 0.947i)12-s + (0.108 − 0.924i)13-s + (−0.676 + 0.580i)14-s + (0.0230 + 0.317i)15-s + (−0.811 + 0.583i)16-s + (0.0711 + 0.0847i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31677 - 0.751245i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31677 - 0.751245i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.14 + 0.832i)T \) |
| 3 | \( 1 + (-1.73 + 0.0244i)T \) |
| good | 5 | \( 1 + (-0.0414 - 0.711i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-0.543 + 2.29i)T + (-6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (-0.518 + 0.788i)T + (-4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.389 + 3.33i)T + (-12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (-0.293 - 0.349i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.44 - 1.21i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (5.49 - 1.30i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (4.29 + 5.76i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-4.14 - 1.24i)T + (25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-6.12 + 1.07i)T + (34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (1.75 - 0.755i)T + (28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (-2.02 + 1.01i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (-1.07 - 3.59i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-4.06 - 7.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.35 + 3.58i)T + (-23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (3.59 + 3.39i)T + (3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (2.22 - 2.99i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (-0.189 - 0.0690i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.37 + 0.863i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (12.5 + 5.41i)T + (54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (1.86 + 0.806i)T + (56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (-1.88 - 5.18i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.615 + 10.5i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−10.27138067421678979903447100961, −9.676171146849528510470664147917, −8.682336456622614774485553669510, −7.81129176680764716837614336735, −7.44568256059677806965540830282, −6.20315715159742580831252208936, −4.33714790345711103129494046494, −3.49885375772841435752603210945, −2.51346771169641551099999302732, −1.10116037362891642861341506984,
1.55499982094118034397774774116, 2.60441642662567237660304440852, 4.27178605025177881169097407969, 5.33221981173744505019453550621, 6.50621657958830169637080126904, 7.35266717006168681039678447782, 8.298086836384821273888744603850, 8.912372309695322712694059568482, 9.464235122682349867532040118114, 10.32164254204416134172060696246
