| L(s) = 1 | + (−1.36 − 0.366i)2-s + (−2.81 + 1.62i)3-s + (1.73 + i)4-s + (4.89 − 1.01i)5-s + (4.44 − 1.19i)6-s + (−3.91 + 1.04i)7-s + (−1.99 − 2i)8-s + (0.800 − 1.38i)9-s + (−7.05 − 0.404i)10-s + (−3.41 − 0.915i)11-s − 6.51·12-s + (1.34 + 12.9i)13-s + 5.72·14-s + (−12.1 + 10.8i)15-s + (1.99 + 3.46i)16-s + (−14.4 + 24.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (−0.939 + 0.542i)3-s + (0.433 + 0.250i)4-s + (0.979 − 0.203i)5-s + (0.741 − 0.198i)6-s + (−0.559 + 0.149i)7-s + (−0.249 − 0.250i)8-s + (0.0889 − 0.153i)9-s + (−0.705 − 0.0404i)10-s + (−0.310 − 0.0832i)11-s − 0.542·12-s + (0.103 + 0.994i)13-s + 0.409·14-s + (−0.810 + 0.722i)15-s + (0.124 + 0.216i)16-s + (−0.848 + 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.231473 + 0.443366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.231473 + 0.443366i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 5 | \( 1 + (-4.89 + 1.01i)T \) |
| 13 | \( 1 + (-1.34 - 12.9i)T \) |
| good | 3 | \( 1 + (2.81 - 1.62i)T + (4.5 - 7.79i)T^{2} \) |
| 7 | \( 1 + (3.91 - 1.04i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (3.41 + 0.915i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (14.4 - 24.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (35.3 - 9.48i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-2.41 - 4.18i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-16.0 - 27.8i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (5.98 + 5.98i)T + 961iT^{2} \) |
| 37 | \( 1 + (-10.3 + 38.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (-4.98 + 18.6i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (9.17 - 15.8i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-32.9 - 32.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 45.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-20.3 - 75.8i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-36.9 + 63.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-22.4 - 6.02i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-94.9 + 25.4i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (71.9 + 71.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 3.01T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-7.10 + 7.10i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-28.7 - 7.70i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (18.2 + 68.0i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−13.10940438821217879107338127899, −12.35298391266128515577825759694, −10.85220851955228836434695595440, −10.56664166670852394097335888659, −9.369305421556597519695488466948, −8.484522762846339506363553472847, −6.55193562137300431771176249975, −5.89290449650492735774248449050, −4.32293790302541631860450977165, −2.07652680824894867330887570268,
0.44589712033851348132837041247, 2.53068725608298649514771882018, 5.18299179236704472558280447201, 6.37508070801415795964148186670, 6.88664212063295622008206599045, 8.513511087658339348882096461467, 9.709920988581544898825060268013, 10.59141585080561214516293091457, 11.49402264987380968397260757387, 12.80481274462426810147505257939
