L(s) = 1 | − 5.43i·3-s − 6.12i·5-s + (6.21 − 3.23i)7-s − 20.5·9-s + 15.2·11-s − 3.00i·13-s − 33.3·15-s + 21.6i·17-s + 11.8i·19-s + (−17.5 − 33.7i)21-s + 1.72·23-s − 12.5·25-s + 62.8i·27-s − 41.1·29-s − 8.50i·31-s + ⋯ |
L(s) = 1 | − 1.81i·3-s − 1.22i·5-s + (0.887 − 0.461i)7-s − 2.28·9-s + 1.38·11-s − 0.231i·13-s − 2.22·15-s + 1.27i·17-s + 0.626i·19-s + (−0.836 − 1.60i)21-s + 0.0748·23-s − 0.502·25-s + 2.32i·27-s − 1.41·29-s − 0.274i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.393568 - 1.60942i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.393568 - 1.60942i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-6.21 + 3.23i)T \) |
good | 3 | \( 1 + 5.43iT - 9T^{2} \) |
| 5 | \( 1 + 6.12iT - 25T^{2} \) |
| 11 | \( 1 - 15.2T + 121T^{2} \) |
| 13 | \( 1 + 3.00iT - 169T^{2} \) |
| 17 | \( 1 - 21.6iT - 289T^{2} \) |
| 19 | \( 1 - 11.8iT - 361T^{2} \) |
| 23 | \( 1 - 1.72T + 529T^{2} \) |
| 29 | \( 1 + 41.1T + 841T^{2} \) |
| 31 | \( 1 + 8.50iT - 961T^{2} \) |
| 37 | \( 1 - 53.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 43.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 36.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 30.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 5.67iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 28.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.53T + 5.04e3T^{2} \) |
| 73 | \( 1 - 94.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 81.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 86.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 27.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 100. iT - 9.40e3T^{2} \) |
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
−11.85913084027110616298108339856, −11.10049987485123803245633908615, −9.309558900876032836657393874734, −8.293927472038911760299587374120, −7.78902103254259643463225995123, −6.55535244885698167774804996329, −5.58360821431261346907296676174, −4.06358808103318717592658499857, −1.76752898517719469551970079677, −1.02799315211862341221752386818,
2.69301788139933848461388439477, 3.88906274554816288996551382748, 4.89024839856780463284394815877, 6.09241155617433872354559315846, 7.43486464086160729526220675946, 9.053930275128304978292592955233, 9.383117443430298472984075685296, 10.65805405723710091554995984560, 11.25641344735848149095518574041, 11.82156018933534916097125905429