| 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 | \( 1 + 71.0T + 9.40e3T^{2} \) |
| 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
−8.135589255732862608779033061227, −7.44145959771594498466550157992, −6.95002112449127280032584011318, −6.42157417241543934486681064267, −5.22446657587713172290956780511, −4.52404363377250284119572419699, −3.41565764907048206573841650878, −2.57957247435821393864585179088, −1.99742424395015442490134663602, −0.04901770613239544445605270364,
1.04930615530866881041453078400, 1.84774137812026125449658379681, 3.25094824145587296244316472227, 4.20734504979965356655009771278, 5.03381896655801017841996547474, 5.44869378892922709195125327540, 6.32804497479521533186112943817, 7.55509159917915859979063104832, 8.216647006873735168141503294939, 8.582766562805136641280192586330
