| L(s) = 1 | + (−1.56 − 0.507i)2-s + (3.90 + 3.90i)3-s + (−1.05 − 0.767i)4-s + (1.95 + 1.42i)5-s + (−4.10 − 8.06i)6-s + (−3.77 − 5.89i)7-s + (5.11 + 7.04i)8-s + 21.4i·9-s + (−2.33 − 3.21i)10-s + (−0.808 + 5.10i)11-s + (−1.12 − 7.11i)12-s + (1.15 − 0.587i)13-s + (2.90 + 11.1i)14-s + (2.08 + 13.1i)15-s + (−2.80 − 8.62i)16-s + (−3.45 + 21.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.780 − 0.253i)2-s + (1.30 + 1.30i)3-s + (−0.264 − 0.191i)4-s + (0.391 + 0.284i)5-s + (−0.684 − 1.34i)6-s + (−0.539 − 0.842i)7-s + (0.639 + 0.880i)8-s + 2.38i·9-s + (−0.233 − 0.321i)10-s + (−0.0735 + 0.464i)11-s + (−0.0939 − 0.593i)12-s + (0.0886 − 0.0451i)13-s + (0.207 + 0.793i)14-s + (0.139 + 0.879i)15-s + (−0.175 − 0.538i)16-s + (−0.203 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.911965 + 1.03156i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.911965 + 1.03156i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (3.77 + 5.89i)T \) |
| 41 | \( 1 + (-5.18 - 40.6i)T \) |
| good | 2 | \( 1 + (1.56 + 0.507i)T + (3.23 + 2.35i)T^{2} \) |
| 3 | \( 1 + (-3.90 - 3.90i)T + 9iT^{2} \) |
| 5 | \( 1 + (-1.95 - 1.42i)T + (7.72 + 23.7i)T^{2} \) |
| 11 | \( 1 + (0.808 - 5.10i)T + (-115. - 37.3i)T^{2} \) |
| 13 | \( 1 + (-1.15 + 0.587i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (3.45 - 21.8i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-29.7 - 15.1i)T + (212. + 292. i)T^{2} \) |
| 23 | \( 1 + (5.77 - 17.7i)T + (-427. - 310. i)T^{2} \) |
| 29 | \( 1 + (41.7 - 6.60i)T + (799. - 259. i)T^{2} \) |
| 31 | \( 1 + (18.6 + 25.7i)T + (-296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-45.4 - 32.9i)T + (423. + 1.30e3i)T^{2} \) |
| 43 | \( 1 + (32.3 + 10.5i)T + (1.49e3 + 1.08e3i)T^{2} \) |
| 47 | \( 1 + (0.912 + 1.79i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-97.2 + 15.4i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-73.5 - 23.8i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (25.9 + 79.9i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (1.09 + 6.93i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-3.04 + 19.2i)T + (-4.79e3 - 1.55e3i)T^{2} \) |
| 73 | \( 1 + 17.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-44.9 + 44.9i)T - 6.24e3iT^{2} \) |
| 83 | \( 1 + 33.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-5.94 + 11.6i)T + (-4.65e3 - 6.40e3i)T^{2} \) |
| 97 | \( 1 + (6.36 - 1.00i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−11.27912457444183893153953012696, −10.31108216462401567335409011236, −9.848162024113090502847111136240, −9.417531529531664718706492792910, −8.240141697778981334759005654388, −7.55170383492203286017639178344, −5.65281858801660521810618188471, −4.30620643602526754231497068435, −3.43070221372228387761115512896, −1.88074414909539058226479141357,
0.78856128245376061198180883948, 2.38966025110750056401883243262, 3.49941176573471164175048311116, 5.55568313473681746431931467470, 6.98770353787495291308527490853, 7.50215894307459223281003390490, 8.650633635478196251776314151376, 9.142839897432968538798125867681, 9.660924335250585596717869966845, 11.57834075878136794142969443540
