| L(s) = 1 | + (1.29 + 0.848i)2-s + (0.152 + 0.354i)4-s + (−0.349 − 0.370i)5-s + (−3.96 + 0.463i)7-s + (0.432 − 2.45i)8-s + (−0.136 − 0.774i)10-s + (−1.09 − 3.66i)11-s + (−0.181 − 3.10i)13-s + (−5.50 − 2.76i)14-s + (3.17 − 3.36i)16-s + (1.41 + 0.513i)17-s + (6.30 − 2.29i)19-s + (0.0778 − 0.180i)20-s + (1.69 − 5.66i)22-s + (−1.17 − 0.137i)23-s + ⋯ |
| L(s) = 1 | + (0.912 + 0.600i)2-s + (0.0764 + 0.177i)4-s + (−0.156 − 0.165i)5-s + (−1.49 + 0.175i)7-s + (0.153 − 0.868i)8-s + (−0.0431 − 0.244i)10-s + (−0.331 − 1.10i)11-s + (−0.0502 − 0.862i)13-s + (−1.47 − 0.738i)14-s + (0.793 − 0.840i)16-s + (0.342 + 0.124i)17-s + (1.44 − 0.526i)19-s + (0.0173 − 0.0403i)20-s + (0.361 − 1.20i)22-s + (−0.245 − 0.0287i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25385 - 0.891891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25385 - 0.891891i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.29 - 0.848i)T + (0.792 + 1.83i)T^{2} \) |
| 5 | \( 1 + (0.349 + 0.370i)T + (-0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (3.96 - 0.463i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (1.09 + 3.66i)T + (-9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (0.181 + 3.10i)T + (-12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (-1.41 - 0.513i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-6.30 + 2.29i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (1.17 + 0.137i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (6.17 - 3.10i)T + (17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (0.0276 - 0.0371i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (2.33 - 1.96i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (4.96 - 3.26i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (-0.231 - 0.0549i)T + (38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (-2.82 - 3.80i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (6.81 + 11.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.400 - 1.33i)T + (-49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (0.124 - 0.288i)T + (-41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (-4.90 - 2.46i)T + (40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (-2.03 - 11.5i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.70 + 15.3i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-11.6 - 7.69i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-0.587 - 0.386i)T + (32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (2.00 - 11.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-8.92 + 9.46i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.04027921671883791455454788097, −9.534974743201309623012434447001, −8.386307780794884907735364972392, −7.35798586946739474871278331126, −6.44392587089160686989988127641, −5.73106584060197232666011093092, −5.04427329384364615022335524032, −3.56487943771476065912798403413, −3.09968287009453208064166035050, −0.58057922962240399209997424250,
2.00960465923601191135010000849, 3.25672748982893899122363383509, 3.82478444229656948492862939104, 4.97923517197411852794173029742, 5.91896794965763266574886765707, 7.09889789050897175833865326470, 7.69921949862425676379992700587, 9.208400096091682662963663396899, 9.737719162925162164225253186371, 10.65644944207629599352161720903
