L(s) = 1 | + (1.34 + 0.776i)5-s + (−2.09 − 3.63i)7-s + (13.7 − 7.91i)11-s + (−4.69 + 8.13i)13-s − 23.9i·17-s + 18.9·19-s + (21.7 + 12.5i)23-s + (−11.2 − 19.5i)25-s + (47.8 − 27.6i)29-s + (−19.5 + 33.9i)31-s − 6.51i·35-s + 19.1·37-s + (2.42 + 1.40i)41-s + (−16.8 − 29.2i)43-s + (13.9 − 8.06i)47-s + ⋯ |
L(s) = 1 | + (0.268 + 0.155i)5-s + (−0.299 − 0.519i)7-s + (1.24 − 0.719i)11-s + (−0.361 + 0.625i)13-s − 1.40i·17-s + 0.998·19-s + (0.947 + 0.546i)23-s + (−0.451 − 0.782i)25-s + (1.65 − 0.952i)29-s + (−0.631 + 1.09i)31-s − 0.186i·35-s + 0.518·37-s + (0.0591 + 0.0341i)41-s + (−0.392 − 0.680i)43-s + (0.297 − 0.171i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.66861 - 0.526113i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66861 - 0.526113i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.34 - 0.776i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (2.09 + 3.63i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-13.7 + 7.91i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (4.69 - 8.13i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 23.9iT - 289T^{2} \) |
| 19 | \( 1 - 18.9T + 361T^{2} \) |
| 23 | \( 1 + (-21.7 - 12.5i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-47.8 + 27.6i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (19.5 - 33.9i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 19.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-2.42 - 1.40i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (16.8 + 29.2i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-13.9 + 8.06i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 53.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-8.59 - 4.96i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-31.3 - 54.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (10.2 - 17.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 110.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (20.3 + 35.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (142. - 82.2i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 13.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-77.1 - 133. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.60268451095069653292438858563, −10.23755461656448406655870354309, −9.491615748251451141140579270036, −8.642025143986050279359334690795, −7.20994841727134734501773752502, −6.62654470084895558468454696500, −5.33682111882148463051987171991, −4.06793845613935937100827739226, −2.84596745672364431325722131405, −0.977685382070878909114443237912,
1.43844649256055550534873133676, 3.03120563758690001492982968720, 4.39032434749212114539504659797, 5.62241890071227764478195854175, 6.55827145392279493657459729967, 7.64989724352412407211772696848, 8.858990318014426502970528432966, 9.533650496607349636218160995382, 10.46774290218884731939973094503, 11.59485789588399066899610755721