| L(s) = 1 | + 4·2-s + 4·4-s + 48·5-s + 72·7-s + 96·8-s + 192·10-s + 596·11-s − 338·13-s + 288·14-s − 176·16-s + 268·17-s + 1.12e3·19-s + 192·20-s + 2.38e3·22-s + 1.76e3·23-s − 3.12e3·25-s − 1.35e3·26-s + 288·28-s + 7.61e3·29-s − 4.16e3·31-s − 5.44e3·32-s + 1.07e3·34-s + 3.45e3·35-s + 1.74e4·37-s + 4.51e3·38-s + 4.60e3·40-s + 2.80e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/8·4-s + 0.858·5-s + 0.555·7-s + 0.530·8-s + 0.607·10-s + 1.48·11-s − 0.554·13-s + 0.392·14-s − 0.171·16-s + 0.224·17-s + 0.716·19-s + 0.107·20-s + 1.05·22-s + 0.696·23-s − 0.999·25-s − 0.392·26-s + 0.0694·28-s + 1.68·29-s − 0.777·31-s − 0.939·32-s + 0.159·34-s + 0.476·35-s + 2.09·37-s + 0.506·38-s + 0.455·40-s + 2.60·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.143788045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.143788045\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{2} T + 3 p^{2} T^{2} - p^{7} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 48 T + 5426 T^{2} - 48 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 72 T + 28134 T^{2} - 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 596 T + 312122 T^{2} - 596 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 268 T + 2228454 T^{2} - 268 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1128 T + 4971870 T^{2} - 1128 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1768 T + 12073598 T^{2} - 1768 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7612 T + 55112798 T^{2} - 7612 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4160 T + 24205318 T^{2} + 4160 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 17468 T + 184139086 T^{2} - 17468 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 28000 T + 412511706 T^{2} - 28000 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 24328 T + 8556274 p T^{2} + 24328 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18108 T + 278422754 T^{2} - 18108 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1420 T + 800695790 T^{2} + 1420 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6788 T - 153114342 T^{2} + 6788 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 37148 T + 1888667614 T^{2} + 37148 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 106112 T + 5372723950 T^{2} - 106112 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 30460 T + 2533993202 T^{2} - 30460 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 37620 T + 1766924310 T^{2} + 37620 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2160 T + 3629418462 T^{2} + 2160 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 207004 T + 18578057034 T^{2} + 207004 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 74136 T + 5425645898 T^{2} - 74136 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 121156 T + 20409714022 T^{2} + 121156 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.80741501117769035926180786198, −12.53318571641340285707309373011, −11.80439357398834173209831221918, −11.19859746543636167053299570459, −11.15915964533860884163289295048, −10.11320140024922410404484256459, −9.499023225714270795034706021445, −9.489164829915041476372706255648, −8.560899438465912616408354077976, −7.904638234245235414156626823013, −7.24450169990620140809014501717, −6.69846984538876081195653790268, −5.97289090439526263402576372378, −5.50978185656620324919853646234, −4.53918896788814584074214742370, −4.42010259669463258918844254286, −3.38757234143922460740790070886, −2.43601461632388723244599581649, −1.62969722632411532817823769363, −0.885320642736629004213960293718,
0.885320642736629004213960293718, 1.62969722632411532817823769363, 2.43601461632388723244599581649, 3.38757234143922460740790070886, 4.42010259669463258918844254286, 4.53918896788814584074214742370, 5.50978185656620324919853646234, 5.97289090439526263402576372378, 6.69846984538876081195653790268, 7.24450169990620140809014501717, 7.904638234245235414156626823013, 8.560899438465912616408354077976, 9.489164829915041476372706255648, 9.499023225714270795034706021445, 10.11320140024922410404484256459, 11.15915964533860884163289295048, 11.19859746543636167053299570459, 11.80439357398834173209831221918, 12.53318571641340285707309373011, 12.80741501117769035926180786198
