| L(s) = 1 | + (0.819 − 0.573i)2-s + (−1.59 − 0.683i)3-s + (0.342 − 0.939i)4-s + (−1.40 − 1.73i)5-s + (−1.69 + 0.352i)6-s + (−3.56 − 0.955i)7-s + (−0.258 − 0.965i)8-s + (2.06 + 2.17i)9-s + (−2.14 − 0.618i)10-s + (−0.294 − 0.170i)11-s + (−1.18 + 1.26i)12-s + (4.18 + 0.366i)13-s + (−3.47 + 1.26i)14-s + (1.04 + 3.72i)15-s + (−0.766 − 0.642i)16-s + (−4.59 + 3.21i)17-s + ⋯ |
| L(s) = 1 | + (0.579 − 0.405i)2-s + (−0.918 − 0.394i)3-s + (0.171 − 0.469i)4-s + (−0.628 − 0.777i)5-s + (−0.692 + 0.143i)6-s + (−1.34 − 0.361i)7-s + (−0.0915 − 0.341i)8-s + (0.688 + 0.725i)9-s + (−0.679 − 0.195i)10-s + (−0.0888 − 0.0513i)11-s + (−0.342 + 0.364i)12-s + (1.16 + 0.101i)13-s + (−0.927 + 0.337i)14-s + (0.270 + 0.962i)15-s + (−0.191 − 0.160i)16-s + (−1.11 + 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.100030 + 0.183516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100030 + 0.183516i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.59 + 0.683i)T \) |
| 5 | \( 1 + (1.40 + 1.73i)T \) |
| 19 | \( 1 + (-1.82 - 3.95i)T \) |
| good | 7 | \( 1 + (3.56 + 0.955i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.294 + 0.170i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.18 - 0.366i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (4.59 - 3.21i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (8.17 - 3.81i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (0.583 + 3.31i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.44 + 4.23i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.23 + 1.23i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.92 + 2.29i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.21 + 2.90i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-5.28 + 7.54i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (6.46 - 3.01i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (0.998 - 5.66i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.81 + 1.02i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (3.62 + 2.53i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (2.31 + 6.35i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.237 - 2.71i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (10.8 - 12.9i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.963 + 0.258i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-7.90 + 6.63i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (5.06 + 7.23i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−10.36870068431005988094744735262, −9.542705397434535804355705045500, −8.315889614320126323675989731130, −7.27593512197085042500089893842, −6.14468672299293831651677040690, −5.74123081392877890660259405869, −4.20660315682446799235254545782, −3.70974268006746112975315543188, −1.69633404543757679195405500250, −0.10726843884141760977148179046,
2.88485609467794877106896638315, 3.79326993390023260104398169669, 4.79312248258254235690987829540, 6.16926047159855705138636394172, 6.44695456430604543601710228887, 7.34967867055239080371939848307, 8.708617610635793520476859537889, 9.676163276239191209180857104582, 10.67839109742617051278645259751, 11.32602168997313980258856640841
