L(s) = 1 | − 0.928i·2-s + 3.13·4-s − 4.06i·5-s − 2.64·7-s − 6.62i·8-s − 3.77·10-s − 9.90i·11-s + 9.26·13-s + 2.45i·14-s + 6.40·16-s + 7.63i·17-s − 13.1·19-s − 12.7i·20-s − 9.19·22-s − 33.2i·23-s + ⋯ |
L(s) = 1 | − 0.464i·2-s + 0.784·4-s − 0.812i·5-s − 0.377·7-s − 0.828i·8-s − 0.377·10-s − 0.900i·11-s + 0.713·13-s + 0.175i·14-s + 0.400·16-s + 0.449i·17-s − 0.690·19-s − 0.637i·20-s − 0.417·22-s − 1.44i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.22275 - 1.22275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22275 - 1.22275i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 2 | \( 1 + 0.928iT - 4T^{2} \) |
| 5 | \( 1 + 4.06iT - 25T^{2} \) |
| 11 | \( 1 + 9.90iT - 121T^{2} \) |
| 13 | \( 1 - 9.26T + 169T^{2} \) |
| 17 | \( 1 - 7.63iT - 289T^{2} \) |
| 19 | \( 1 + 13.1T + 361T^{2} \) |
| 23 | \( 1 + 33.2iT - 529T^{2} \) |
| 29 | \( 1 - 12.8iT - 841T^{2} \) |
| 31 | \( 1 - 22.1T + 961T^{2} \) |
| 37 | \( 1 + 56.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 56.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 83.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 39.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 93.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 11.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 39.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 44.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 110. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−12.24137239691939695991816284391, −10.98830898228759053425686360861, −10.44339057998220889566277296698, −9.025470066333639664138847444979, −8.233584314592992775856148007216, −6.72012880682844456471029668710, −5.84999921228652272429517724302, −4.19192259538390413989128578290, −2.82224312678982177941555818626, −1.08255699745915902361352106185,
2.14664810008875027767639423945, 3.54329925873822736092938473837, 5.37546610529332607986084618053, 6.61170547756635686402607463262, 7.13442396247840795112385888768, 8.321442919246827842186050997019, 9.728992056154674149365663026210, 10.68898663611234763846979965973, 11.49888517218669100513756903391, 12.49351103959219522675999135276