| L(s) = 1 | − 1.73i·3-s − 5.37i·5-s − 2.99·9-s + 8.59·11-s + 21.0i·13-s − 9.31·15-s + 5.48i·17-s − 7.24i·19-s − 28.0·23-s − 3.94·25-s + 5.19i·27-s + 40.3·29-s − 40.5i·31-s − 14.8i·33-s + 66.6·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.07i·5-s − 0.333·9-s + 0.781·11-s + 1.61i·13-s − 0.621·15-s + 0.322i·17-s − 0.381i·19-s − 1.21·23-s − 0.157·25-s + 0.192i·27-s + 1.39·29-s − 1.30i·31-s − 0.450i·33-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.875523490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875523490\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5.37iT - 25T^{2} \) |
| 11 | \( 1 - 8.59T + 121T^{2} \) |
| 13 | \( 1 - 21.0iT - 169T^{2} \) |
| 17 | \( 1 - 5.48iT - 289T^{2} \) |
| 19 | \( 1 + 7.24iT - 361T^{2} \) |
| 23 | \( 1 + 28.0T + 529T^{2} \) |
| 29 | \( 1 - 40.3T + 841T^{2} \) |
| 31 | \( 1 + 40.5iT - 961T^{2} \) |
| 37 | \( 1 - 66.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 0.932T + 1.84e3T^{2} \) |
| 47 | \( 1 + 85.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 44.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 63.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 32.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 47.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 14.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 140. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 122.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 33.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 36.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.2iT - 9.40e3T^{2} \) |
| 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
−8.462798692850724764233382157058, −8.009218777111366901908296194728, −6.82591653002765110130185355305, −6.41144087300109482324738097347, −5.43661008187426002220741379886, −4.41995590106419038099623614154, −3.94156639907255030520475809003, −2.36953952090554538756808384462, −1.54874242258754765082910830679, −0.52159861196825195571800730790,
1.06238817232626185232729214296, 2.68495994639364673023581082729, 3.17361903309831327944712097200, 4.15231587142254722730625915026, 5.05955500481776240766289848072, 6.12564924367275457041494468599, 6.49143220533009756811548215130, 7.69150703638462955215506849139, 8.118339086094108408337426353045, 9.194692614520535680445572423276
