L(s) = 1 | − 9.50i·5-s + 2.36·7-s − 12.8i·11-s − 15.9·13-s + 14.0i·17-s + 6.58·19-s + 26.3i·23-s − 65.3·25-s − 1.52i·29-s − 3.57·31-s − 22.4i·35-s + 22.1·37-s + 48.9i·41-s − 83.8·43-s − 35.5i·47-s + ⋯ |
L(s) = 1 | − 1.90i·5-s + 0.337·7-s − 1.17i·11-s − 1.22·13-s + 0.828i·17-s + 0.346·19-s + 1.14i·23-s − 2.61·25-s − 0.0524i·29-s − 0.115·31-s − 0.641i·35-s + 0.599·37-s + 1.19i·41-s − 1.95·43-s − 0.755i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2087864166\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2087864166\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 9.50iT - 25T^{2} \) |
| 7 | \( 1 - 2.36T + 49T^{2} \) |
| 11 | \( 1 + 12.8iT - 121T^{2} \) |
| 13 | \( 1 + 15.9T + 169T^{2} \) |
| 17 | \( 1 - 14.0iT - 289T^{2} \) |
| 19 | \( 1 - 6.58T + 361T^{2} \) |
| 23 | \( 1 - 26.3iT - 529T^{2} \) |
| 29 | \( 1 + 1.52iT - 841T^{2} \) |
| 31 | \( 1 + 3.57T + 961T^{2} \) |
| 37 | \( 1 - 22.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 48.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 83.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 35.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 65.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 94.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 82.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 95.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 33.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 11.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 10.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 71.0T + 9.40e3T^{2} \) |
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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
−8.582766562805136641280192586330, −8.216647006873735168141503294939, −7.55509159917915859979063104832, −6.32804497479521533186112943817, −5.44869378892922709195125327540, −5.03381896655801017841996547474, −4.20734504979965356655009771278, −3.25094824145587296244316472227, −1.84774137812026125449658379681, −1.04930615530866881041453078400,
0.04901770613239544445605270364, 1.99742424395015442490134663602, 2.57957247435821393864585179088, 3.41565764907048206573841650878, 4.52404363377250284119572419699, 5.22446657587713172290956780511, 6.42157417241543934486681064267, 6.95002112449127280032584011318, 7.44145959771594498466550157992, 8.135589255732862608779033061227