| L(s) = 1 | + (0.366 − 1.36i)2-s + (−1.73 − i)4-s + (−4.41 − 4.41i)5-s + (2.11 + 7.89i)7-s + (−2 + 1.99i)8-s + (−7.64 + 4.41i)10-s + (−18.5 − 4.96i)11-s + (−12.9 − 1.42i)13-s + 11.5·14-s + (1.99 + 3.46i)16-s + (19.9 + 11.5i)17-s + (0.417 − 0.111i)19-s + (3.23 + 12.0i)20-s + (−13.5 + 23.4i)22-s + (−36.4 + 21.0i)23-s + ⋯ |
| L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.433 − 0.250i)4-s + (−0.882 − 0.882i)5-s + (0.302 + 1.12i)7-s + (−0.250 + 0.249i)8-s + (−0.764 + 0.441i)10-s + (−1.68 − 0.451i)11-s + (−0.993 − 0.109i)13-s + 0.825·14-s + (0.124 + 0.216i)16-s + (1.17 + 0.679i)17-s + (0.0219 − 0.00588i)19-s + (0.161 + 0.602i)20-s + (−0.616 + 1.06i)22-s + (−1.58 + 0.915i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0432916 + 0.0947865i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0432916 + 0.0947865i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (12.9 + 1.42i)T \) |
| good | 5 | \( 1 + (4.41 + 4.41i)T + 25iT^{2} \) |
| 7 | \( 1 + (-2.11 - 7.89i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (18.5 + 4.96i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (-19.9 - 11.5i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-0.417 + 0.111i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (36.4 - 21.0i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-3.25 - 5.64i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (17.8 + 17.8i)T + 961iT^{2} \) |
| 37 | \( 1 + (1.37 + 0.369i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-10.8 + 40.3i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (51.1 + 29.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (15.0 - 15.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 8.90T + 2.80e3T^{2} \) |
| 59 | \( 1 + (11.4 + 42.6i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-44.8 + 77.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-10.2 + 38.2i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (8.56 - 2.29i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (5.92 - 5.92i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 115.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-34.2 - 34.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (58.0 + 15.5i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (129. - 34.6i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−11.62279984593902688107047093861, −10.45084754408385940588995244209, −9.485126556146236897404720608359, −8.197464634169711554571204928722, −7.891941531330935388493415698383, −5.60336474071645936206175529108, −5.07350987667052339845050116569, −3.59773109009902639756617329749, −2.16708264167266790440724954009, −0.04982759118664190443963266800,
2.88204582195715461706363044044, 4.21599018506570631844014197412, 5.24796412773882211001969438364, 6.84383655296221502850524157432, 7.64891326104398015341952563862, 7.979680808314934059712766505872, 9.957371023390319990924409093169, 10.44584463804462958791271403489, 11.65405922331961720003561813989, 12.59552718102631492752088146309
