| L(s) = 1 | − 3.70i·5-s − 0.419·7-s − 4.94i·11-s + 4.77·13-s − 23.6i·17-s + 27.0·19-s + 21.0i·23-s + 11.2·25-s − 3.30i·29-s + 17.3·31-s + 1.55i·35-s − 49.9·37-s − 40.5i·41-s + 65.1·43-s + 27.3i·47-s + ⋯ |
| L(s) = 1 | − 0.740i·5-s − 0.0599·7-s − 0.449i·11-s + 0.367·13-s − 1.39i·17-s + 1.42·19-s + 0.916i·23-s + 0.451·25-s − 0.113i·29-s + 0.560·31-s + 0.0443i·35-s − 1.34·37-s − 0.988i·41-s + 1.51·43-s + 0.582i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.066520344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066520344\) |
| \(L(2)\) |
|
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 | \( 1 + 3.70iT - 25T^{2} \) |
| 7 | \( 1 + 0.419T + 49T^{2} \) |
| 11 | \( 1 + 4.94iT - 121T^{2} \) |
| 13 | \( 1 - 4.77T + 169T^{2} \) |
| 17 | \( 1 + 23.6iT - 289T^{2} \) |
| 19 | \( 1 - 27.0T + 361T^{2} \) |
| 23 | \( 1 - 21.0iT - 529T^{2} \) |
| 29 | \( 1 + 3.30iT - 841T^{2} \) |
| 31 | \( 1 - 17.3T + 961T^{2} \) |
| 37 | \( 1 + 49.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 40.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 65.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 27.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 59.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 49.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 69.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 73.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7.63T + 5.32e3T^{2} \) |
| 79 | \( 1 + 129.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 68.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 159. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.59T + 9.40e3T^{2} \) |
| 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
−8.520087379276206599044926004313, −7.56456155230587335924384195776, −7.07672735995080832560208120250, −5.92299022943253468086863471581, −5.30915465228536613036907540635, −4.61439358004014945661367604492, −3.52984489790456168802714645111, −2.77271272436935071885369304247, −1.38684364583485050130095390830, −0.55490350946408258079658244156,
1.06464004722394258849848654759, 2.20431054607989310237337670634, 3.17853892633376299906151294486, 3.90843002134005003399827981686, 4.92825183777919657882516074537, 5.77890943098217542264345467078, 6.65508794033445209749638163766, 7.08080721679898197625570736995, 8.127285283114017904942052722685, 8.591399145884116935666493298188
