| L(s) = 1 | + (2.44 − 1.73i)3-s + (−4.5 − 2.59i)5-s + (−4.17 − 7.22i)7-s + (2.99 − 8.48i)9-s + (−0.825 + 0.476i)11-s + (4.84 − 8.39i)13-s + (−15.5 + 1.43i)15-s + 18.8i·17-s + 24.6·19-s + (−22.7 − 10.4i)21-s + (−0.825 − 0.476i)23-s + (1 + 1.73i)25-s + (−7.34 − 25.9i)27-s + (11.8 − 6.84i)29-s + (1.52 − 2.63i)31-s + ⋯ |
| L(s) = 1 | + (0.816 − 0.577i)3-s + (−0.900 − 0.519i)5-s + (−0.596 − 1.03i)7-s + (0.333 − 0.942i)9-s + (−0.0750 + 0.0433i)11-s + (0.372 − 0.645i)13-s + (−1.03 + 0.0953i)15-s + 1.11i·17-s + 1.29·19-s + (−1.08 − 0.499i)21-s + (−0.0359 − 0.0207i)23-s + (0.0400 + 0.0692i)25-s + (−0.272 − 0.962i)27-s + (0.408 − 0.235i)29-s + (0.0491 − 0.0850i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.932982 - 1.10152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.932982 - 1.10152i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.44 + 1.73i)T \) |
| good | 5 | \( 1 + (4.5 + 2.59i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (4.17 + 7.22i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (0.825 - 0.476i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-4.84 + 8.39i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 18.8iT - 289T^{2} \) |
| 19 | \( 1 - 24.6T + 361T^{2} \) |
| 23 | \( 1 + (0.825 + 0.476i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-11.8 + 6.84i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-1.52 + 2.63i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 46.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (9.45 + 5.45i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-22.5 - 39.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (39.2 - 22.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 94.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.2 - 9.39i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (6.54 + 11.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-37.5 + 64.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 18.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7.90T + 5.32e3T^{2} \) |
| 79 | \( 1 + (21.8 + 37.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-112. + 65.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-54.9 - 95.1i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.79239715338053513303606419253, −11.80731733809370779077921578571, −10.46093851124041623403978193574, −9.360792969584319585403220389594, −8.086147724185144064846346225606, −7.57719767346730460336213020282, −6.24914464636957940769422695232, −4.24640654739287124804145749730, −3.23886559272518401240426636071, −0.925777670434345044508533533931,
2.69396914812535854176709091210, 3.70185427456299874378854740897, 5.22030790239369887826294440978, 6.89168188208284409090052918462, 7.983251700254422522710477324063, 9.088942613404189964053084332492, 9.821571962588950084096131406275, 11.23798371017051866979734676943, 11.94766131098231632520478243979, 13.29474629475097127843237991864
