L(s) = 1 | + (−2.89 + 0.795i)3-s + (0.355 + 0.615i)5-s + (−2.70 − 1.56i)7-s + (7.73 − 4.60i)9-s + (−14.3 − 8.30i)11-s + (−9.17 − 15.8i)13-s + (−1.51 − 1.49i)15-s − 9.69·17-s − 8.20i·19-s + (9.06 + 2.36i)21-s + (−1.94 + 1.12i)23-s + (12.2 − 21.2i)25-s + (−18.7 + 19.4i)27-s + (−20.8 + 36.0i)29-s + (21.6 − 12.4i)31-s + ⋯ |
L(s) = 1 | + (−0.964 + 0.265i)3-s + (0.0710 + 0.123i)5-s + (−0.386 − 0.223i)7-s + (0.859 − 0.511i)9-s + (−1.30 − 0.754i)11-s + (−0.705 − 1.22i)13-s + (−0.101 − 0.0998i)15-s − 0.570·17-s − 0.431i·19-s + (0.431 + 0.112i)21-s + (−0.0847 + 0.0489i)23-s + (0.489 − 0.848i)25-s + (−0.692 + 0.721i)27-s + (−0.717 + 1.24i)29-s + (0.697 − 0.402i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.173229 - 0.365046i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.173229 - 0.365046i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.89 - 0.795i)T \) |
good | 5 | \( 1 + (-0.355 - 0.615i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (2.70 + 1.56i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (14.3 + 8.30i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (9.17 + 15.8i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 9.69T + 289T^{2} \) |
| 19 | \( 1 + 8.20iT - 361T^{2} \) |
| 23 | \( 1 + (1.94 - 1.12i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (20.8 - 36.0i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-21.6 + 12.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 40.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-25.6 - 44.5i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-56.6 - 32.7i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (29.2 + 16.9i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 90.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-66.2 + 38.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-1.35 + 2.35i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-34.5 + 19.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 38.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (94.4 + 54.5i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (113. + 65.5i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 38.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + (12.1 - 21.1i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.68931482653758839729845985942, −11.26183199783884385197956166615, −10.56179961949646815493488580482, −9.748083310279368623113135246878, −8.193945644806655585583075939827, −6.96525510693214129453765604461, −5.76263742681022596609106488481, −4.80089977219270124886536227260, −3.01493985709193450447813414433, −0.27976211106785359599932910644,
2.15315996536543920158095248299, 4.45045011868136400471517243094, 5.50474316594636149366239532205, 6.76335239060465201638265713165, 7.67398595010991033858246072140, 9.304264040088293858492440427824, 10.24906784114971227346531925752, 11.27015268063859955110112982961, 12.32348591575673771003923760807, 12.91562845739669768984714839945