| L(s) = 1 | + (−1.24 + 1.48i)2-s + (2.80 − 1.06i)3-s + (0.0409 + 0.232i)4-s + (−1.47 + 4.06i)5-s + (−1.91 + 5.49i)6-s + (1.54 − 8.75i)7-s + (−7.11 − 4.10i)8-s + (6.73 − 5.96i)9-s + (−4.19 − 7.26i)10-s + (−2.52 − 6.93i)11-s + (0.361 + 0.607i)12-s + (−12.4 + 10.4i)13-s + (11.0 + 13.2i)14-s + (0.172 + 12.9i)15-s + (14.0 − 5.13i)16-s + (9.06 − 5.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.623 + 0.743i)2-s + (0.935 − 0.354i)3-s + (0.0102 + 0.0580i)4-s + (−0.295 + 0.812i)5-s + (−0.319 + 0.915i)6-s + (0.220 − 1.25i)7-s + (−0.889 − 0.513i)8-s + (0.748 − 0.662i)9-s + (−0.419 − 0.726i)10-s + (−0.229 − 0.630i)11-s + (0.0301 + 0.0506i)12-s + (−0.955 + 0.801i)13-s + (0.791 + 0.943i)14-s + (0.0115 + 0.864i)15-s + (0.881 − 0.320i)16-s + (0.533 − 0.307i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.798964 + 0.363758i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.798964 + 0.363758i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.80 + 1.06i)T \) |
| good | 2 | \( 1 + (1.24 - 1.48i)T + (-0.694 - 3.93i)T^{2} \) |
| 5 | \( 1 + (1.47 - 4.06i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-1.54 + 8.75i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (2.52 + 6.93i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (12.4 - 10.4i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-9.06 + 5.23i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (8.63 - 14.9i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-12.9 + 2.28i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (26.8 - 32.0i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-6.43 - 36.5i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (31.9 + 55.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (16.3 + 19.4i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (2.32 - 0.847i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-45.9 - 8.10i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 - 50.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-1.43 + 3.93i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-6.31 + 35.8i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-13.2 + 11.1i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (51.5 - 29.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-26.3 + 45.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (1.19 + 1.00i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-72.6 + 86.6i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-41.4 - 23.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-34.5 + 12.5i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−17.23008015115645219313110772351, −16.21890267728341876000211166857, −14.76801801157460369098177724512, −14.03930241605059907942608792414, −12.41726274489347159324094294811, −10.55395330161629531680170497459, −8.988780601341764660930544113409, −7.52872186638377858738170413465, −6.99597141332820283215351208685, −3.48571863997888107698436769117,
2.44049824280711758748949063170, 5.11322502024610515615236086004, 8.107089582127252118690262106510, 9.127676877878353177936012708210, 10.12864252927773773600527379065, 11.84067567080231810791038076091, 12.93354514442402018075380563798, 15.01862226668426082931821082414, 15.28627724032906076055560983391, 17.18044686798748664332214887759
