L(s) = 1 | − 2.71i·2-s − 3.35·4-s − 7.17i·5-s + 2.64·7-s − 1.75i·8-s − 19.4·10-s + 2.71i·11-s + 6.35·13-s − 7.17i·14-s − 18.1·16-s + 6.38i·17-s − 30.5·19-s + 24.0i·20-s + 7.35·22-s − 27.9i·23-s + ⋯ |
L(s) = 1 | − 1.35i·2-s − 0.838·4-s − 1.43i·5-s + 0.377·7-s − 0.218i·8-s − 1.94·10-s + 0.246i·11-s + 0.488·13-s − 0.512i·14-s − 1.13·16-s + 0.375i·17-s − 1.60·19-s + 1.20i·20-s + 0.334·22-s − 1.21i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.43999i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43999i\) |
\(L(2)\) |
|
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 | \( 1 + 2.71iT - 4T^{2} \) |
| 5 | \( 1 + 7.17iT - 25T^{2} \) |
| 11 | \( 1 - 2.71iT - 121T^{2} \) |
| 13 | \( 1 - 6.35T + 169T^{2} \) |
| 17 | \( 1 - 6.38iT - 289T^{2} \) |
| 19 | \( 1 + 30.5T + 361T^{2} \) |
| 23 | \( 1 + 27.9iT - 529T^{2} \) |
| 29 | \( 1 - 37.0iT - 841T^{2} \) |
| 31 | \( 1 - 44.3T + 961T^{2} \) |
| 37 | \( 1 - 71.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 4.68T + 1.84e3T^{2} \) |
| 47 | \( 1 + 65.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 36.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 45.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 93.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 80.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 20.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 94.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 52.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 98.2T + 9.40e3T^{2} \) |
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
−11.92825338834154841466193264477, −10.92940580717194441243710500061, −10.08605976518209848770528808472, −8.898147393268865166219460007823, −8.310782518013089284413698638405, −6.47630001991422402188450904474, −4.84653302238203726524711875246, −4.02936669833723122438795236836, −2.20247004354222222976786371787, −0.879714582107235026578769306463,
2.60293449140380309263588697661, 4.34240232919681319874182734348, 5.95181494010663835341818846769, 6.52410439288956774032694763610, 7.59914572549747203251965824758, 8.347952608025344022939181259992, 9.745073113974419101645705576085, 10.98640995386590556495198412568, 11.53764317046855470659468613967, 13.31423876686308478108256872919