| L(s) = 1 | + (−0.709 − 1.22i)2-s + (1.68 + 0.420i)3-s + (−0.994 + 1.73i)4-s + (−2.09 + 2.49i)5-s + (−0.676 − 2.35i)6-s + (−2.63 + 1.51i)7-s + (2.82 − 0.0135i)8-s + (2.64 + 1.41i)9-s + (4.54 + 0.794i)10-s + (−0.326 + 0.565i)11-s + (−2.40 + 2.49i)12-s + (0.487 + 2.76i)13-s + (3.72 + 2.14i)14-s + (−4.57 + 3.31i)15-s + (−2.02 − 3.45i)16-s + (1.31 + 3.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.501 − 0.865i)2-s + (0.970 + 0.242i)3-s + (−0.497 + 0.867i)4-s + (−0.937 + 1.11i)5-s + (−0.276 − 0.961i)6-s + (−0.994 + 0.574i)7-s + (0.999 − 0.00477i)8-s + (0.881 + 0.471i)9-s + (1.43 + 0.251i)10-s + (−0.0984 + 0.170i)11-s + (−0.693 + 0.720i)12-s + (0.135 + 0.767i)13-s + (0.995 + 0.572i)14-s + (−1.18 + 0.856i)15-s + (−0.505 − 0.862i)16-s + (0.319 + 0.876i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.817603 + 0.415812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.817603 + 0.415812i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.709 + 1.22i)T \) |
| 3 | \( 1 + (-1.68 - 0.420i)T \) |
| 19 | \( 1 + (1.88 + 3.93i)T \) |
| good | 5 | \( 1 + (2.09 - 2.49i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.63 - 1.51i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.326 - 0.565i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.487 - 2.76i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.31 - 3.61i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.40 + 4.53i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.771 - 2.11i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.94 - 1.70i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.89T + 37T^{2} \) |
| 41 | \( 1 + (-10.3 - 1.83i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.58 + 9.03i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (1.83 + 0.668i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (5.81 + 6.93i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (8.69 - 3.16i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (4.97 - 4.17i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (2.95 - 8.12i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.41 - 3.70i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.0225 + 0.127i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-13.9 - 2.45i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.37 - 2.38i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.45 + 0.256i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.69 + 1.34i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−12.41954556586800285370417953974, −11.10195404520059933857981666527, −10.54592438905988959891371076716, −9.371118878058976377740119039791, −8.741337084232360295101907881513, −7.56815223786022657035224926480, −6.71981112917271675209570226569, −4.31216024658148732807474656638, −3.32091030076652752022904453660, −2.48169667231875465113599452023,
0.843478444686760049959142111052, 3.46319212517277962212311426957, 4.64211387560332165071262795836, 6.14116303334255833426212660331, 7.68452638355749028266600731210, 7.74400242476139247960873023015, 9.125419385592094125765252075861, 9.553384310537164632637397133641, 10.88003426003025302311218860387, 12.53000692212876150742862816465
