| L(s) = 1 | + 9i·5-s + 5·7-s + 117i·11-s − 34·13-s + 450i·17-s − 64·19-s + 612i·23-s + 544·25-s + 1.06e3i·29-s − 697·31-s + 45i·35-s − 748·37-s + 684i·41-s + 2.61e3·43-s − 2.64e3i·47-s + ⋯ |
| L(s) = 1 | + 0.359i·5-s + 0.102·7-s + 0.966i·11-s − 0.201·13-s + 1.55i·17-s − 0.177·19-s + 1.15i·23-s + 0.870·25-s + 1.26i·29-s − 0.725·31-s + 0.0367i·35-s − 0.546·37-s + 0.406i·41-s + 1.41·43-s − 1.19i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.00366 + 1.00366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00366 + 1.00366i\) |
| \(L(3)\) |
|
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 | \( 1 - 9iT - 625T^{2} \) |
| 7 | \( 1 - 5T + 2.40e3T^{2} \) |
| 11 | \( 1 - 117iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 34T + 2.85e4T^{2} \) |
| 17 | \( 1 - 450iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 64T + 1.30e5T^{2} \) |
| 23 | \( 1 - 612iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.06e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 697T + 9.23e5T^{2} \) |
| 37 | \( 1 + 748T + 1.87e6T^{2} \) |
| 41 | \( 1 - 684iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.61e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.64e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.07e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.81e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.40e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.21e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 6.57e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.51e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 7.50e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 5.48e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.87e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.05e4T + 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.09186791230056260538149114567, −12.33158767946432614357884831771, −11.03801273603851871844645563982, −10.16218332862058325924040514815, −8.957264686496322542703138345278, −7.65873495575248042996558133761, −6.58311622632324317087726498629, −5.12205023863242180547277956679, −3.61446601895199997040693208283, −1.81074364198028364053065974691,
0.64334920823501279540036670758, 2.74604489179466577702223320029, 4.47663938633016616264341739330, 5.76164196829385482110821900163, 7.15524687000509825256436040542, 8.441811738788672930349170847216, 9.379555639978207914235101469964, 10.70007531978504615614967744152, 11.67421060643095213104317458864, 12.73343990958904782252015097480
