| L(s) = 1 | + (−0.909 + 1.08i)2-s + (2.80 − 1.05i)3-s + (−0.347 − 1.96i)4-s + (2.86 − 7.86i)5-s + (−1.41 + 4.00i)6-s + (−1.95 + 11.0i)7-s + (2.44 + 1.41i)8-s + (6.77 − 5.92i)9-s + (5.92 + 10.2i)10-s + (0.538 + 1.47i)11-s + (−3.05 − 5.16i)12-s + (−6.82 + 5.72i)13-s + (−10.2 − 12.1i)14-s + (−0.256 − 25.1i)15-s + (−3.75 + 1.36i)16-s + (−16.9 + 9.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.936 − 0.351i)3-s + (−0.0868 − 0.492i)4-s + (0.572 − 1.57i)5-s + (−0.235 + 0.666i)6-s + (−0.278 + 1.58i)7-s + (0.306 + 0.176i)8-s + (0.752 − 0.658i)9-s + (0.592 + 1.02i)10-s + (0.0489 + 0.134i)11-s + (−0.254 − 0.430i)12-s + (−0.524 + 0.440i)13-s + (−0.729 − 0.869i)14-s + (−0.0170 − 1.67i)15-s + (−0.234 + 0.0855i)16-s + (−0.999 + 0.576i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0275i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.23893 - 0.0170708i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23893 - 0.0170708i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 - 1.08i)T \) |
| 3 | \( 1 + (-2.80 + 1.05i)T \) |
| good | 5 | \( 1 + (-2.86 + 7.86i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (1.95 - 11.0i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-0.538 - 1.47i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (6.82 - 5.72i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (16.9 - 9.80i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-4.86 + 8.42i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (13.8 - 2.43i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-7.05 + 8.40i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-8.04 - 45.6i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-7.89 - 13.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (6.59 + 7.86i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-24.6 + 8.96i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (42.2 + 7.45i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 - 41.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-20.6 + 56.8i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-13.2 + 75.3i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-2.09 + 1.75i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-65.1 + 37.6i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.0 + 57.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-38.8 - 32.5i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-21.4 + 25.5i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (75.2 + 43.4i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-132. + 48.2i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−15.34727899979336858470042607137, −14.02470376208640458952901207506, −12.89269698857207770685129155601, −12.12121394989362781938603140809, −9.638380765067485238328347486290, −8.980459455829226804845130196739, −8.276768219358936406297735997951, −6.39669096328354072461246350859, −4.92466492004446814686200434021, −2.01037972240035355407577531342,
2.60090867671788618794889814631, 3.93046818787463394335535337215, 6.84862986048166972930601249553, 7.78219801954984622527198079300, 9.665271963985792970321897803697, 10.26303971772962283868866827106, 11.13692501951564503005521558647, 13.25368964158125296624844262834, 13.94185056886573556779850959125, 14.84968456408918991132237872304
