| L(s) = 1 | − 4.38i·5-s + 3.19·13-s + 0.138i·17-s − 14.1·25-s − 8.62i·29-s − 9.39·37-s − 1.41i·41-s + 7·49-s − 7.07i·53-s − 15.3·61-s − 14.0i·65-s + 2.80·73-s + 0.607·85-s + 12.8i·89-s + 8·97-s + ⋯ |
| L(s) = 1 | − 1.95i·5-s + 0.886·13-s + 0.0336i·17-s − 2.83·25-s − 1.60i·29-s − 1.54·37-s − 0.220i·41-s + 49-s − 0.971i·53-s − 1.97·61-s − 1.73i·65-s + 0.328·73-s + 0.0659·85-s + 1.36i·89-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182440075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182440075\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4.38iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 - 0.138iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.62iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2.80T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 12.8iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
| 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
−8.133228802893642992439818046001, −7.27669146665147551003662306293, −6.17635504860274576368857966193, −5.63927154070331116753211035818, −4.89273390192100788110266274406, −4.23092302942697922284976631043, −3.53247572780904680312551956047, −2.10956459053755008440120579878, −1.28159682950827740947064428673, −0.31683963366681137534308895202,
1.52243726918188566356411789082, 2.53877794328209561406103702826, 3.32509242252547628569842056586, 3.77821495235021711603103446633, 4.96753054076817479872080553494, 5.97869190528821748623176885630, 6.39449352062673386901484677338, 7.20683803835320606932085021379, 7.54647918355352121884610719728, 8.588818363928517167135499381770
