| L(s) = 1 | + 84·5-s + 193·7-s − 348·11-s + 845·13-s + 1.69e3·17-s + 79·19-s + 564·23-s + 3.93e3·25-s + 6.43e3·29-s − 4.94e3·31-s + 1.62e4·35-s − 3.80e3·37-s − 1.24e4·41-s + 4.93e3·43-s − 8.12e3·47-s + 2.04e4·49-s − 3.31e4·53-s − 2.92e4·55-s − 4.24e4·59-s − 1.78e4·61-s + 7.09e4·65-s + 6.76e4·67-s − 2.81e4·71-s − 1.39e4·73-s − 6.71e4·77-s + 8.39e4·79-s + 3.33e4·83-s + ⋯ |
| L(s) = 1 | + 1.50·5-s + 1.48·7-s − 0.867·11-s + 1.38·13-s + 1.41·17-s + 0.0502·19-s + 0.222·23-s + 1.25·25-s + 1.42·29-s − 0.923·31-s + 2.23·35-s − 0.456·37-s − 1.15·41-s + 0.407·43-s − 0.536·47-s + 1.21·49-s − 1.62·53-s − 1.30·55-s − 1.58·59-s − 0.613·61-s + 2.08·65-s + 1.84·67-s − 0.662·71-s − 0.306·73-s − 1.29·77-s + 1.51·79-s + 0.531·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.903809910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.903809910\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 84 T + p^{5} T^{2} \) |
| 7 | \( 1 - 193 T + p^{5} T^{2} \) |
| 11 | \( 1 + 348 T + p^{5} T^{2} \) |
| 13 | \( 1 - 5 p^{2} T + p^{5} T^{2} \) |
| 17 | \( 1 - 1692 T + p^{5} T^{2} \) |
| 19 | \( 1 - 79 T + p^{5} T^{2} \) |
| 23 | \( 1 - 564 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6432 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4940 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3805 T + p^{5} T^{2} \) |
| 41 | \( 1 + 12480 T + p^{5} T^{2} \) |
| 43 | \( 1 - 4936 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8124 T + p^{5} T^{2} \) |
| 53 | \( 1 + 33192 T + p^{5} T^{2} \) |
| 59 | \( 1 + 42492 T + p^{5} T^{2} \) |
| 61 | \( 1 + 17833 T + p^{5} T^{2} \) |
| 67 | \( 1 - 67699 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28152 T + p^{5} T^{2} \) |
| 73 | \( 1 + 13975 T + p^{5} T^{2} \) |
| 79 | \( 1 - 83983 T + p^{5} T^{2} \) |
| 83 | \( 1 - 33384 T + p^{5} T^{2} \) |
| 89 | \( 1 - 77868 T + p^{5} T^{2} \) |
| 97 | \( 1 + 2083 T + p^{5} T^{2} \) |
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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
−10.48146232248368097183442812509, −9.496293199221324077831629713122, −8.461179947244253089000903938073, −7.79179108137017633229700816377, −6.39437829057779106449712251060, −5.49823371358969809388136639546, −4.87439327828248663384459323136, −3.21883692509193413407900782736, −1.87610632519781724420805003185, −1.17520362394749769998616154000,
1.17520362394749769998616154000, 1.87610632519781724420805003185, 3.21883692509193413407900782736, 4.87439327828248663384459323136, 5.49823371358969809388136639546, 6.39437829057779106449712251060, 7.79179108137017633229700816377, 8.461179947244253089000903938073, 9.496293199221324077831629713122, 10.48146232248368097183442812509
