| 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
−17.46072013446407250098600019962, −15.51091639086990433695649872811, −14.25947165488612370124488946487, −13.53205631997774705133214755512, −11.44875376958992246024649542604, −10.81846541663677018439771637081, −9.088643622524829271271600601183, −6.87615320080910192965153021905, −6.06307413647121780568827760655, −2.24960862630413174051912956637,
3.99778883427206642462520436874, 5.77221167186882453221502391571, 8.293069921214032223450690307573, 9.123191937857592511813552675435, 10.76692240834658611505665929639, 12.39229955576414426166924951594, 13.38359417497739028581714084844, 15.32274356385907501354259025964, 16.35549861357028339763021860279, 16.98731242429170735378434701116
