| L(s) = 1 | + (−0.119 − 1.99i)2-s + (2.95 − 0.536i)3-s + (−3.97 + 0.479i)4-s + (3.50 − 3.57i)5-s + (−1.42 − 5.82i)6-s + (−0.568 + 0.568i)7-s + (1.43 + 7.87i)8-s + (8.42 − 3.16i)9-s + (−7.54 − 6.55i)10-s + (−10.1 − 7.40i)11-s + (−11.4 + 3.54i)12-s + (2.42 − 15.2i)13-s + (1.20 + 1.06i)14-s + (8.41 − 12.4i)15-s + (15.5 − 3.80i)16-s + (8.27 + 16.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0599 − 0.998i)2-s + (0.983 − 0.178i)3-s + (−0.992 + 0.119i)4-s + (0.700 − 0.714i)5-s + (−0.237 − 0.971i)6-s + (−0.0811 + 0.0811i)7-s + (0.179 + 0.983i)8-s + (0.935 − 0.352i)9-s + (−0.754 − 0.655i)10-s + (−0.926 − 0.672i)11-s + (−0.955 + 0.295i)12-s + (0.186 − 1.17i)13-s + (0.0858 + 0.0761i)14-s + (0.560 − 0.827i)15-s + (0.971 − 0.237i)16-s + (0.486 + 0.955i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.836859 - 1.91900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.836859 - 1.91900i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.119 + 1.99i)T \) |
| 3 | \( 1 + (-2.95 + 0.536i)T \) |
| 5 | \( 1 + (-3.50 + 3.57i)T \) |
| good | 7 | \( 1 + (0.568 - 0.568i)T - 49iT^{2} \) |
| 11 | \( 1 + (10.1 + 7.40i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-2.42 + 15.2i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-8.27 - 16.2i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-6.43 + 19.8i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (23.4 - 3.71i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-5.03 - 15.5i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-1.43 - 0.465i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-15.9 - 2.51i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-23.9 - 32.9i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (11.9 + 11.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (32.5 - 63.8i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-27.6 + 54.2i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (31.2 + 42.9i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (2.14 + 1.55i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-4.46 - 8.76i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-22.9 - 70.6i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-64.7 + 10.2i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-25.5 - 78.6i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-28.3 - 55.6i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-130. - 95.0i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-8.19 + 16.0i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.00849485060820089909837391732, −10.10933487863904397036433445718, −9.431412783708041453149243419108, −8.358709843489054253183443052683, −7.952238391120545056343660955496, −5.94341938237842754120868133981, −4.86074706290835193884770012898, −3.42663865512616634518916475346, −2.43508772677163498974974541894, −1.00450129519833997836200578994,
2.09355726789093485871599999525, 3.61221574047493586121345256854, 4.85406572778052663993532583967, 6.11179853781501621105577172650, 7.21186043271477119293813401969, 7.82313744661634654396102338311, 9.033048378191302740007865561973, 9.858196985146777443382931503815, 10.32606041669448420822179469326, 12.06606195736977261780122766233
