| L(s) = 1 | + (1.95 − 2.27i)3-s + (−3.25 − 8.93i)5-s + (−0.410 − 2.32i)7-s + (−1.38 − 8.89i)9-s + (−4.40 + 12.1i)11-s + (−12.2 − 10.2i)13-s + (−26.6 − 10.0i)15-s + (12.2 + 7.04i)17-s + (3.29 + 5.70i)19-s + (−6.09 − 3.60i)21-s + (25.9 + 4.58i)23-s + (−50.0 + 41.9i)25-s + (−22.9 − 14.2i)27-s + (0.977 + 1.16i)29-s + (0.620 − 3.52i)31-s + ⋯ |
| L(s) = 1 | + (0.650 − 0.759i)3-s + (−0.650 − 1.78i)5-s + (−0.0585 − 0.332i)7-s + (−0.153 − 0.988i)9-s + (−0.400 + 1.10i)11-s + (−0.939 − 0.788i)13-s + (−1.77 − 0.668i)15-s + (0.717 + 0.414i)17-s + (0.173 + 0.300i)19-s + (−0.290 − 0.171i)21-s + (1.12 + 0.199i)23-s + (−2.00 + 1.67i)25-s + (−0.850 − 0.526i)27-s + (0.0337 + 0.0401i)29-s + (0.0200 − 0.113i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0274i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0179018 - 1.30615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0179018 - 1.30615i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.95 + 2.27i)T \) |
| good | 5 | \( 1 + (3.25 + 8.93i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (0.410 + 2.32i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (4.40 - 12.1i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (12.2 + 10.2i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-12.2 - 7.04i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-3.29 - 5.70i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-25.9 - 4.58i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-0.977 - 1.16i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-0.620 + 3.52i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (11.0 - 19.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-31.4 + 37.5i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (78.8 + 28.6i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-34.9 + 6.16i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 65.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-17.2 - 47.3i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (11.5 + 65.6i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (72.5 + 60.8i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (71.8 + 41.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (47.8 + 82.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-25.9 + 21.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (7.14 + 8.50i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (8.83 - 5.10i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-66.7 - 24.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−10.27072411193675170017370510516, −9.392679035038538142523694808709, −8.541318925429675069207630934613, −7.73634230834947699759155610914, −7.20856356083028924688893460353, −5.49157372960833189315097956781, −4.65052373249047964538278937498, −3.41646548647335937232113935562, −1.75558494539444886514246264903, −0.50200491375893038970072114367,
2.70921433543980940067115423869, 3.09189497733562812471768219681, 4.34601089093557038750159930029, 5.67707915939127367230804955089, 6.97787835911657845408237244783, 7.64450210116531796057693865285, 8.695993061315468149483451330557, 9.707489871991557678837261604051, 10.52140523311832960978965983336, 11.21520029839046513907137040403
