| L(s) = 1 | + (−1.02 − 0.978i)2-s + (−0.991 + 1.42i)3-s + (0.0861 + 1.99i)4-s + (0.499 − 0.471i)5-s + (2.40 − 0.481i)6-s + (−3.08 + 0.360i)7-s + (1.86 − 2.12i)8-s + (−1.03 − 2.81i)9-s + (−0.970 − 0.00732i)10-s + (1.31 − 0.394i)11-s + (−2.92 − 1.85i)12-s + (4.56 − 0.265i)13-s + (3.50 + 2.64i)14-s + (0.174 + 1.17i)15-s + (−3.98 + 0.344i)16-s + (4.36 + 1.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.722 − 0.691i)2-s + (−0.572 + 0.820i)3-s + (0.0430 + 0.999i)4-s + (0.223 − 0.210i)5-s + (0.980 − 0.196i)6-s + (−1.16 + 0.136i)7-s + (0.659 − 0.751i)8-s + (−0.345 − 0.938i)9-s + (−0.307 − 0.00231i)10-s + (0.397 − 0.118i)11-s + (−0.844 − 0.536i)12-s + (1.26 − 0.0736i)13-s + (0.936 + 0.708i)14-s + (0.0450 + 0.303i)15-s + (−0.996 + 0.0860i)16-s + (1.05 + 0.385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.668775 + 0.274605i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.668775 + 0.274605i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 + 0.978i)T \) |
| 3 | \( 1 + (0.991 - 1.42i)T \) |
| good | 5 | \( 1 + (-0.499 + 0.471i)T + (0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (3.08 - 0.360i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-1.31 + 0.394i)T + (9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-4.56 + 0.265i)T + (12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-4.36 - 1.58i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (1.77 + 4.87i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (1.14 + 0.134i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (-4.54 - 9.05i)T + (-17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (6.18 - 8.31i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-0.0755 - 0.0900i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (4.06 - 2.67i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (-0.380 + 1.60i)T + (-38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-3.75 - 5.04i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (2.32 - 1.34i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.92 - 2.97i)T + (49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-8.34 - 3.59i)T + (41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (0.789 - 1.57i)T + (-40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (-0.622 - 3.52i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.76 + 15.6i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-8.91 - 5.86i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-4.91 + 7.46i)T + (-32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (0.656 - 3.72i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (4.72 - 5.01i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.62534164106293779251662993427, −9.876553781612540830790606680608, −9.014298587476492135189895525082, −8.663850369926457222329601113939, −7.03111227117972497889003402641, −6.25306168809495394802762390436, −5.11272101502478000267975902039, −3.68892348190606041222071299812, −3.19190153440873064216299131853, −1.14642658717947980473967754109,
0.65152186124853597510435520930, 2.12770331000132729265129401327, 3.88502033851688663042905815962, 5.60155960996782918853933105508, 6.18675002913826550963535187520, 6.72884112428573816048904815837, 7.79144596339192750182639260382, 8.470521414651504305430286495349, 9.748846702416562018728775262850, 10.16261865174479299709830989471
