| L(s) = 1 | + (1.45 + 1.95i)2-s + (−1.12 + 3.76i)4-s + (2.46 − 1.62i)5-s + (−1.57 + 1.66i)7-s + (−4.42 + 1.61i)8-s + (6.75 + 2.45i)10-s + (−2.11 + 1.06i)11-s + (0.243 + 0.563i)13-s + (−5.54 − 0.647i)14-s + (−3.01 − 1.98i)16-s + (5.54 − 4.64i)17-s + (−3.45 − 2.89i)19-s + (3.32 + 11.1i)20-s + (−5.15 − 2.58i)22-s + (−5.13 − 5.44i)23-s + ⋯ |
| L(s) = 1 | + (1.02 + 1.38i)2-s + (−0.564 + 1.88i)4-s + (1.10 − 0.725i)5-s + (−0.593 + 0.629i)7-s + (−1.56 + 0.569i)8-s + (2.13 + 0.777i)10-s + (−0.638 + 0.320i)11-s + (0.0674 + 0.156i)13-s + (−1.48 − 0.173i)14-s + (−0.753 − 0.495i)16-s + (1.34 − 1.12i)17-s + (−0.792 − 0.664i)19-s + (0.744 + 2.48i)20-s + (−1.09 − 0.552i)22-s + (−1.07 − 1.13i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29662 + 1.71218i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29662 + 1.71218i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.45 - 1.95i)T + (-0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (-2.46 + 1.62i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (1.57 - 1.66i)T + (-0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (2.11 - 1.06i)T + (6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (-0.243 - 0.563i)T + (-8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (-5.54 + 4.64i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (3.45 + 2.89i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (5.13 + 5.44i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (2.15 - 0.252i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-3.02 + 0.716i)T + (27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (-0.740 - 4.19i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-3.84 + 5.16i)T + (-11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.236 + 4.06i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-0.841 - 0.199i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (-1.55 - 2.69i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.548 + 0.275i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-1.56 - 5.23i)T + (-50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (0.280 + 0.0327i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (7.71 + 2.80i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (14.3 - 5.21i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-4.45 - 5.97i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-3.27 - 4.39i)T + (-23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (8.64 - 3.14i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.42 - 0.940i)T + (38.4 + 89.0i)T^{2} \) |
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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
−12.68321447732375532448112993522, −12.10618508923617940698651709090, −10.21230295911745294512221884147, −9.251066984896539190203727927705, −8.290461886964331941258732680133, −7.11542087388109518959242767876, −6.01841057553353408175661246871, −5.44589623851022930082539906932, −4.44081095709144284956755140919, −2.66842601747331721558411226166,
1.74301188183236085992019553816, 3.04779711718619457545330751737, 3.96633649853468067928297075469, 5.62776414045979908668151158246, 6.16334463111499671023569138501, 7.84671121413312898580794743185, 9.693478725015719903427166412062, 10.24567755655949182611437432358, 10.72327941963398274579497088587, 11.92313605914980549089584322664
