| L(s) = 1 | + (−0.374 + 0.445i)2-s + (−0.428 + 2.96i)3-s + (0.635 + 3.60i)4-s + (2.62 − 7.20i)5-s + (−1.16 − 1.30i)6-s + (0.231 − 1.31i)7-s + (−3.86 − 2.22i)8-s + (−8.63 − 2.54i)9-s + (2.23 + 3.86i)10-s + (0.367 + 1.01i)11-s + (−10.9 + 0.343i)12-s + (16.0 − 13.4i)13-s + (0.498 + 0.594i)14-s + (20.2 + 10.8i)15-s + (−11.3 + 4.12i)16-s + (−12.1 + 7.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.187 + 0.222i)2-s + (−0.142 + 0.989i)3-s + (0.158 + 0.901i)4-s + (0.524 − 1.44i)5-s + (−0.193 − 0.216i)6-s + (0.0330 − 0.187i)7-s + (−0.482 − 0.278i)8-s + (−0.959 − 0.282i)9-s + (0.223 + 0.386i)10-s + (0.0334 + 0.0918i)11-s + (−0.914 + 0.0286i)12-s + (1.23 − 1.03i)13-s + (0.0356 + 0.0424i)14-s + (1.35 + 0.724i)15-s + (−0.707 + 0.257i)16-s + (−0.716 + 0.413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.804550 + 0.392828i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.804550 + 0.392828i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.428 - 2.96i)T \) |
| good | 2 | \( 1 + (0.374 - 0.445i)T + (-0.694 - 3.93i)T^{2} \) |
| 5 | \( 1 + (-2.62 + 7.20i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-0.231 + 1.31i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-0.367 - 1.01i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-16.0 + 13.4i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (12.1 - 7.03i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (9.79 - 16.9i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (1.76 - 0.311i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (25.8 - 30.7i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (2.95 + 16.7i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-1.80 - 3.11i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (3.09 + 3.68i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-16.1 + 5.87i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-45.1 - 7.95i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + 51.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (32.0 - 88.0i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (3.88 - 22.0i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (14.9 - 12.5i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-74.9 + 43.2i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-18.0 + 31.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (22.3 + 18.7i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (76.1 - 90.8i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-104. - 60.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-9.05 + 3.29i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−16.98731242429170735378434701116, −16.35549861357028339763021860279, −15.32274356385907501354259025964, −13.38359417497739028581714084844, −12.39229955576414426166924951594, −10.76692240834658611505665929639, −9.123191937857592511813552675435, −8.293069921214032223450690307573, −5.77221167186882453221502391571, −3.99778883427206642462520436874,
2.24960862630413174051912956637, 6.06307413647121780568827760655, 6.87615320080910192965153021905, 9.088643622524829271271600601183, 10.81846541663677018439771637081, 11.44875376958992246024649542604, 13.53205631997774705133214755512, 14.25947165488612370124488946487, 15.51091639086990433695649872811, 17.46072013446407250098600019962
