| L(s) = 1 | + (1.16 + 0.323i)2-s + (−1.51 + 0.847i)3-s + (−0.467 − 0.282i)4-s + (−1.06 + 0.0411i)5-s + (−2.02 + 0.495i)6-s + (−5.09 − 0.796i)7-s + (−2.10 − 2.23i)8-s + (1.56 − 2.55i)9-s + (−1.24 − 0.295i)10-s + (0.460 + 3.36i)11-s + (0.945 + 0.0301i)12-s + (1.07 − 1.56i)13-s + (−5.65 − 2.57i)14-s + (1.56 − 0.960i)15-s + (−1.21 − 2.30i)16-s + (−1.45 + 0.730i)17-s + ⋯ |
| L(s) = 1 | + (0.821 + 0.228i)2-s + (−0.872 + 0.489i)3-s + (−0.233 − 0.141i)4-s + (−0.474 + 0.0183i)5-s + (−0.828 + 0.202i)6-s + (−1.92 − 0.301i)7-s + (−0.744 − 0.789i)8-s + (0.521 − 0.853i)9-s + (−0.393 − 0.0932i)10-s + (0.138 + 1.01i)11-s + (0.272 + 0.00869i)12-s + (0.298 − 0.434i)13-s + (−1.51 − 0.687i)14-s + (0.404 − 0.247i)15-s + (−0.304 − 0.577i)16-s + (−0.353 + 0.177i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00559952 - 0.0257023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00559952 - 0.0257023i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.51 - 0.847i)T \) |
| good | 2 | \( 1 + (-1.16 - 0.323i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (1.06 - 0.0411i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (5.09 + 0.796i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.460 - 3.36i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 1.56i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (1.45 - 0.730i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.362 - 6.23i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (-1.67 + 4.33i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (5.05 + 3.61i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (3.36 - 3.85i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (3.97 + 5.33i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-1.40 + 5.44i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-3.72 - 3.38i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (3.71 + 4.25i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (0.178 - 1.01i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.85 + 3.20i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (4.86 - 2.93i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (2.72 - 1.94i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (4.36 + 14.5i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (-3.76 + 0.892i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (9.19 - 9.01i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (1.47 + 5.72i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (0.365 - 1.22i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (11.0 + 0.427i)T + (96.7 + 7.51i)T^{2} \) |
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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
−12.15646683637033202411368030860, −10.58680779076145733142842752456, −9.902682051051234784445199180389, −9.155394624653621692316411198062, −7.21446451726203288329480353888, −6.32377367636543231160688282600, −5.55940717536934434120449947224, −4.12047797066952878053440153929, −3.58493765285315313709239916170, −0.01740433108513981311048839488,
2.94379481694835715410245447105, 3.98368631536296277494245483054, 5.43046125550602553755720945624, 6.24172698350541842386880248285, 7.21324694593640383289917158648, 8.757019354725940136431467790798, 9.601616628943135117962276838205, 11.18370800848804992912470287026, 11.63865420582129746157481161075, 12.68965963013065126134077430625
