| L(s) = 1 | + (−1.82 − 1.93i)2-s + (−0.293 + 5.04i)4-s + (−2.82 + 0.330i)5-s + (1.70 + 0.858i)7-s + (6.21 − 5.21i)8-s + (5.79 + 4.86i)10-s + (0.733 + 1.70i)11-s + (−0.131 − 0.0311i)13-s + (−1.45 − 4.86i)14-s + (−11.3 − 1.32i)16-s + (−0.193 + 1.09i)17-s + (−1.05 − 5.97i)19-s + (−0.836 − 14.3i)20-s + (1.94 − 4.51i)22-s + (−0.416 + 0.209i)23-s + ⋯ |
| L(s) = 1 | + (−1.28 − 1.36i)2-s + (−0.146 + 2.52i)4-s + (−1.26 + 0.147i)5-s + (0.646 + 0.324i)7-s + (2.19 − 1.84i)8-s + (1.83 + 1.53i)10-s + (0.221 + 0.512i)11-s + (−0.0364 − 0.00862i)13-s + (−0.389 − 1.30i)14-s + (−2.83 − 0.331i)16-s + (−0.0470 + 0.266i)17-s + (−0.241 − 1.37i)19-s + (−0.187 − 3.21i)20-s + (0.415 − 0.963i)22-s + (−0.0869 + 0.0436i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0140276 + 0.0272175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0140276 + 0.0272175i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.82 + 1.93i)T + (-0.116 + 1.99i)T^{2} \) |
| 5 | \( 1 + (2.82 - 0.330i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (-1.70 - 0.858i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-0.733 - 1.70i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (0.131 + 0.0311i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (0.193 - 1.09i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (1.05 + 5.97i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (0.416 - 0.209i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (-0.125 + 0.420i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (2.36 + 1.55i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (7.77 - 2.82i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-3.81 + 4.04i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (4.47 - 6.01i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (7.12 - 4.68i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (-2.94 + 5.09i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.58 - 12.9i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (0.446 + 7.66i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (3.24 + 10.8i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (0.604 + 0.506i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (2.76 - 2.32i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.76 - 9.29i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (11.4 + 12.1i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (8.34 - 7.00i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (2.28 + 0.267i)T + (94.3 + 22.3i)T^{2} \) |
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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
−9.878628429444050247701772070550, −9.014896279626991078604492909360, −8.323215743131466944676541085416, −7.66172586214596326298377919450, −6.84700876671715531615219508370, −4.79339049982651612848745469651, −3.89441921122725145159284818311, −2.86358146115055200115314424708, −1.65524974367554403019138806287, −0.02616014746882858283242238773,
1.44812074542342406519397171073, 3.78566536414404686306820919691, 4.91353870057884672784692227481, 5.89875673094698694584925202919, 6.95354902410517117348589692844, 7.64964813995866927118441398317, 8.272759577248543885611790823883, 8.799564855337591523253658667840, 9.909366220390176139343262171499, 10.71784464144288716092992848450
