| L(s) = 1 | − 5·2-s − 7·4-s + 25·5-s − 244·7-s + 195·8-s − 125·10-s − 794·11-s − 169·13-s + 1.22e3·14-s − 751·16-s + 1.53e3·17-s + 2.70e3·19-s − 175·20-s + 3.97e3·22-s + 702·23-s + 625·25-s + 845·26-s + 1.70e3·28-s + 5.03e3·29-s − 3.63e3·31-s − 2.48e3·32-s − 7.67e3·34-s − 6.10e3·35-s − 7.05e3·37-s − 1.35e4·38-s + 4.87e3·40-s + 294·41-s + ⋯ |
| L(s) = 1 | − 0.883·2-s − 0.218·4-s + 0.447·5-s − 1.88·7-s + 1.07·8-s − 0.395·10-s − 1.97·11-s − 0.277·13-s + 1.66·14-s − 0.733·16-s + 1.28·17-s + 1.71·19-s − 0.0978·20-s + 1.74·22-s + 0.276·23-s + 1/5·25-s + 0.245·26-s + 0.411·28-s + 1.11·29-s − 0.679·31-s − 0.428·32-s − 1.13·34-s − 0.841·35-s − 0.847·37-s − 1.51·38-s + 0.481·40-s + 0.0273·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + 5 T + p^{5} T^{2} \) |
| 7 | \( 1 + 244 T + p^{5} T^{2} \) |
| 11 | \( 1 + 794 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1534 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2706 T + p^{5} T^{2} \) |
| 23 | \( 1 - 702 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5038 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3634 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7058 T + p^{5} T^{2} \) |
| 41 | \( 1 - 294 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7618 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3020 T + p^{5} T^{2} \) |
| 53 | \( 1 + 626 T + p^{5} T^{2} \) |
| 59 | \( 1 - 30066 T + p^{5} T^{2} \) |
| 61 | \( 1 + 5806 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12436 T + p^{5} T^{2} \) |
| 71 | \( 1 + 4734 T + p^{5} T^{2} \) |
| 73 | \( 1 + 14694 T + p^{5} T^{2} \) |
| 79 | \( 1 + 39804 T + p^{5} T^{2} \) |
| 83 | \( 1 - 41776 T + p^{5} T^{2} \) |
| 89 | \( 1 + 7970 T + p^{5} T^{2} \) |
| 97 | \( 1 + 78050 T + p^{5} T^{2} \) |
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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
−9.743902646471741272122261448815, −8.789863080363405286551356528908, −7.68950830140404416187585011146, −7.12274396930424926953079812703, −5.74597537600469411337019960216, −5.09282144664350691403144810578, −3.42043726312884599581113219166, −2.63877058162349065955732485083, −0.933295351507904149325908147535, 0,
0.933295351507904149325908147535, 2.63877058162349065955732485083, 3.42043726312884599581113219166, 5.09282144664350691403144810578, 5.74597537600469411337019960216, 7.12274396930424926953079812703, 7.68950830140404416187585011146, 8.789863080363405286551356528908, 9.743902646471741272122261448815
