| L(s) = 1 | + (193. + 111. i)2-s + (1.67e4 + 2.89e4i)4-s + (2.00e4 − 1.15e4i)5-s + (3.61e5 − 6.26e5i)7-s + 3.80e6i·8-s + 5.17e6·10-s + (2.12e7 + 1.22e7i)11-s + (5.32e7 + 9.21e7i)13-s + (1.39e8 − 8.06e7i)14-s + (−1.50e8 + 2.61e8i)16-s + 1.48e8i·17-s − 1.39e9·19-s + (6.70e8 + 3.87e8i)20-s + (2.73e9 + 4.74e9i)22-s + (4.34e9 − 2.51e9i)23-s + ⋯ |
| L(s) = 1 | + (1.51 + 0.871i)2-s + (1.02 + 1.76i)4-s + (0.256 − 0.148i)5-s + (0.439 − 0.760i)7-s + 1.81i·8-s + 0.517·10-s + (1.09 + 0.629i)11-s + (0.848 + 1.46i)13-s + (1.32 − 0.765i)14-s + (−0.561 + 0.972i)16-s + 0.362i·17-s − 1.56·19-s + (0.523 + 0.302i)20-s + (1.09 + 1.90i)22-s + (1.27 − 0.737i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000829 - 0.999i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.000829 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(5.652719356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.652719356\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-193. - 111. i)T + (8.19e3 + 1.41e4i)T^{2} \) |
| 5 | \( 1 + (-2.00e4 + 1.15e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 7 | \( 1 + (-3.61e5 + 6.26e5i)T + (-3.39e11 - 5.87e11i)T^{2} \) |
| 11 | \( 1 + (-2.12e7 - 1.22e7i)T + (1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + (-5.32e7 - 9.21e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 17 | \( 1 - 1.48e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 1.39e9T + 7.99e17T^{2} \) |
| 23 | \( 1 + (-4.34e9 + 2.51e9i)T + (5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + (-1.35e9 - 7.80e8i)T + (1.48e20 + 2.57e20i)T^{2} \) |
| 31 | \( 1 + (3.09e9 + 5.35e9i)T + (-3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 + 8.97e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + (4.32e10 - 2.49e10i)T + (1.89e22 - 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-1.16e11 + 2.01e11i)T + (-3.69e22 - 6.39e22i)T^{2} \) |
| 47 | \( 1 + (-3.23e11 - 1.86e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 + 6.91e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (-1.64e10 + 9.47e9i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (7.88e11 - 1.36e12i)T + (-4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (4.27e12 + 7.40e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 + 1.08e13iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 8.40e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + (-7.09e11 + 1.22e12i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 + (1.70e13 + 9.83e12i)T + (3.68e26 + 6.37e26i)T^{2} \) |
| 89 | \( 1 + 1.03e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-4.39e13 + 7.61e13i)T + (-3.26e27 - 5.65e27i)T^{2} \) |
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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
−14.34051443999413925400383016678, −13.49297711361317979450262334411, −12.33050055023492475990613702589, −11.02394682035649637128422755097, −8.869110096134515051888248679798, −7.11138503249532038811012131916, −6.29370627825219428908816101396, −4.59446113861362216703375167532, −3.86728516629453956025094304988, −1.70944526570634834014992396811,
1.18554047250004112640033991540, 2.58719763863260943136604769511, 3.79440576275935628464674966932, 5.32697924535893431277203357614, 6.28709217468612525760549751049, 8.678292644419885433201735057055, 10.55107457124594760836833925902, 11.47578789193623873036850471226, 12.59701396963037500275940968380, 13.62179862193246936374454734431
