| L(s) = 1 | + (−0.819 + 0.573i)2-s + (1.27 − 1.16i)3-s + (0.342 − 0.939i)4-s + (−1.35 − 1.77i)5-s + (−0.376 + 1.69i)6-s + (0.943 + 0.252i)7-s + (0.258 + 0.965i)8-s + (0.265 − 2.98i)9-s + (2.13 + 0.674i)10-s + (−4.69 − 2.71i)11-s + (−0.661 − 1.60i)12-s + (−3.15 − 0.275i)13-s + (−0.917 + 0.333i)14-s + (−3.81 − 0.678i)15-s + (−0.766 − 0.642i)16-s + (0.681 − 0.477i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 + 0.405i)2-s + (0.737 − 0.675i)3-s + (0.171 − 0.469i)4-s + (−0.608 − 0.793i)5-s + (−0.153 + 0.690i)6-s + (0.356 + 0.0955i)7-s + (0.0915 + 0.341i)8-s + (0.0885 − 0.996i)9-s + (0.674 + 0.213i)10-s + (−1.41 − 0.817i)11-s + (−0.191 − 0.462i)12-s + (−0.874 − 0.0765i)13-s + (−0.245 + 0.0892i)14-s + (−0.984 − 0.175i)15-s + (−0.191 − 0.160i)16-s + (0.165 − 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.375517 - 0.771671i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375517 - 0.771671i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 - 0.573i)T \) |
| 3 | \( 1 + (-1.27 + 1.16i)T \) |
| 5 | \( 1 + (1.35 + 1.77i)T \) |
| 19 | \( 1 + (-4.08 - 1.51i)T \) |
| good | 7 | \( 1 + (-0.943 - 0.252i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (4.69 + 2.71i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.15 + 0.275i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.681 + 0.477i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (4.11 - 1.92i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (0.154 + 0.875i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.15 + 3.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.54 + 6.54i)T - 37iT^{2} \) |
| 41 | \( 1 + (7.34 - 8.74i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.85 + 0.862i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.36 + 6.22i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-11.1 + 5.20i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (-0.414 + 2.34i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.34 + 0.853i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.15 - 6.40i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (5.27 + 14.5i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (0.359 + 4.10i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (-2.85 + 3.40i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.66 - 2.32i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-9.51 + 7.98i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (2.90 + 4.15i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−10.12953445234197792731495913615, −9.352127920677383717990510995129, −8.349519220530181397481640471383, −7.87325840494160365644571401823, −7.34301057424198003060295045084, −5.86020339781941965759136372189, −5.00843623981674146949415726721, −3.47680597032013694186024738641, −2.12135593885586271840547021585, −0.51264487297914865348090824612,
2.27073253152251660434188272755, 3.03640633279827467073393328678, 4.25588806214262464157128897902, 5.21227183112880452108158670939, 7.08190079279914049947013105980, 7.69228968473144358938455493742, 8.314598372122061363164835011541, 9.527034310997649082195096889771, 10.22800829964393041325510049240, 10.68924249405931231963564374148
