| L(s) = 1 | + (−1.94 + 0.976i)2-s + (1.63 − 2.19i)4-s + (0.185 − 0.619i)5-s + (−0.724 + 1.67i)7-s + (−0.277 + 1.57i)8-s + (0.244 + 1.38i)10-s + (1.17 + 0.279i)11-s + (4.45 − 2.93i)13-s + (−0.231 − 3.97i)14-s + (0.571 + 1.90i)16-s + (6.43 + 2.34i)17-s + (−5.97 + 2.17i)19-s + (−1.05 − 1.41i)20-s + (−2.56 + 0.607i)22-s + (1.23 + 2.86i)23-s + ⋯ |
| L(s) = 1 | + (−1.37 + 0.690i)2-s + (0.816 − 1.09i)4-s + (0.0829 − 0.276i)5-s + (−0.273 + 0.634i)7-s + (−0.0980 + 0.556i)8-s + (0.0772 + 0.438i)10-s + (0.354 + 0.0841i)11-s + (1.23 − 0.812i)13-s + (−0.0618 − 1.06i)14-s + (0.142 + 0.477i)16-s + (1.56 + 0.567i)17-s + (−1.37 + 0.499i)19-s + (−0.236 − 0.317i)20-s + (−0.546 + 0.129i)22-s + (0.257 + 0.596i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.569726 + 0.337767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.569726 + 0.337767i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.94 - 0.976i)T + (1.19 - 1.60i)T^{2} \) |
| 5 | \( 1 + (-0.185 + 0.619i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (0.724 - 1.67i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 0.279i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (-4.45 + 2.93i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-6.43 - 2.34i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (5.97 - 2.17i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-1.23 - 2.86i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (0.342 - 5.87i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (-2.75 - 0.321i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (1.09 - 0.918i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (0.996 + 0.500i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (-6.61 + 7.00i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-6.06 + 0.708i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (4.26 + 7.38i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.03 - 0.481i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (2.14 + 2.87i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (0.0709 + 1.21i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (1.41 + 8.02i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.11 + 6.32i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (12.7 - 6.39i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (6.00 - 3.01i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (-2.70 + 15.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.06 - 3.57i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−12.32528009858708589035360107693, −10.83847689216885574381423260030, −10.20014819449548280480132214053, −9.032173397922233691999989493322, −8.541057637341009308829424566085, −7.58119419039053379038968197233, −6.33990559193138981926143480325, −5.57748862132412013281361490154, −3.54944812791804241559511587311, −1.33851800678293230141729355966,
1.04701247386371398371008912518, 2.73891270536811640288532767493, 4.20322948078900561455644565115, 6.17437675320186649912814569724, 7.21868149804442934739495425568, 8.335876651319408789822140560305, 9.127179725941233070829436728975, 10.09206150327335728171759583802, 10.78952563609008950163507203708, 11.54878983044249886745649534702
