| L(s) = 1 | + 4·2-s + 18·3-s + 4·4-s + 72·6-s − 72·7-s + 96·8-s + 243·9-s − 596·11-s + 72·12-s + 338·13-s − 288·14-s − 176·16-s + 268·17-s + 972·18-s + 1.12e3·19-s − 1.29e3·21-s − 2.38e3·22-s + 1.76e3·23-s + 1.72e3·24-s + 1.35e3·26-s + 2.91e3·27-s − 288·28-s − 7.61e3·29-s − 4.16e3·31-s − 5.44e3·32-s − 1.07e4·33-s + 1.07e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/8·4-s + 0.816·6-s − 0.555·7-s + 0.530·8-s + 9-s − 1.48·11-s + 0.144·12-s + 0.554·13-s − 0.392·14-s − 0.171·16-s + 0.224·17-s + 0.707·18-s + 0.716·19-s − 0.641·21-s − 1.05·22-s + 0.696·23-s + 0.612·24-s + 0.392·26-s + 0.769·27-s − 0.0694·28-s − 1.68·29-s − 0.777·31-s − 0.939·32-s − 1.71·33-s + 0.159·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{2} T + 3 p^{2} T^{2} - p^{7} T^{3} + p^{10} T^{4} \) |
| 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
−8.853362523375411720096891327265, −8.847942062503719478108054968186, −8.257394393564681690120802121382, −7.60604456637800196065958990791, −7.43042402226182721456152049657, −7.13287786505171042571074995643, −6.66128351744578133972212491701, −5.84648695083420590029345385543, −5.46642461960274584302184171573, −5.25663775727685170421995918706, −4.40854047344103653240698593672, −4.34016549294829566938022938578, −3.36331666063448486618573521714, −3.31905322588906733435974949742, −2.95430473999340418697243901722, −2.05504434653879142442494511449, −1.80550904378984041155886197547, −1.19184181590593341847169516253, 0, 0,
1.19184181590593341847169516253, 1.80550904378984041155886197547, 2.05504434653879142442494511449, 2.95430473999340418697243901722, 3.31905322588906733435974949742, 3.36331666063448486618573521714, 4.34016549294829566938022938578, 4.40854047344103653240698593672, 5.25663775727685170421995918706, 5.46642461960274584302184171573, 5.84648695083420590029345385543, 6.66128351744578133972212491701, 7.13287786505171042571074995643, 7.43042402226182721456152049657, 7.60604456637800196065958990791, 8.257394393564681690120802121382, 8.847942062503719478108054968186, 8.853362523375411720096891327265
