| L(s) = 1 | − 6·3-s + 10·5-s − 14·7-s + 27·9-s + 16·11-s − 12·13-s − 60·15-s − 124·17-s + 32·19-s + 84·21-s + 272·23-s + 75·25-s − 108·27-s − 132·29-s − 280·31-s − 96·33-s − 140·35-s − 292·37-s + 72·39-s − 508·41-s + 456·43-s + 270·45-s + 1.05e3·47-s + 147·49-s + 744·51-s + 52·53-s + 160·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 0.438·11-s − 0.256·13-s − 1.03·15-s − 1.76·17-s + 0.386·19-s + 0.872·21-s + 2.46·23-s + 3/5·25-s − 0.769·27-s − 0.845·29-s − 1.62·31-s − 0.506·33-s − 0.676·35-s − 1.29·37-s + 0.295·39-s − 1.93·41-s + 1.61·43-s + 0.894·45-s + 3.27·47-s + 3/7·49-s + 2.04·51-s + 0.134·53-s + 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 16 T + 1190 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12 T + 2030 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 11270 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32 T - 4842 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 272 T + 1858 p T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 132 T + 2350 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 280 T + 57582 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 292 T + 80286 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 508 T + 200822 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 456 T + 194774 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1056 T + 485566 T^{2} - 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 52 T + 1798 p T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 440 T + 455702 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 756 T + 479246 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 248 T + 6306 p T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 488 T + 753758 T^{2} + 488 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1092 T + 826454 T^{2} + 1092 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1232 T + 1241118 T^{2} + 1232 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 776 T + 584102 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 156 T + 1340758 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 156 T + 1036454 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.333587887996429382294698081494, −9.292169009989587146818572595639, −8.884763345598069156319389002724, −8.779546121922510928690102687875, −7.43550692007863997360746483486, −7.40761989811732388374900332511, −6.82572561928684254092992612251, −6.70384028344584700557302357391, −5.99730234451416221997723952991, −5.69762026080722578143651562477, −5.20819866167320386195214415906, −4.91358242474775315525359673757, −4.06025702827235363212895873806, −3.90347600751386956696627429261, −2.75399160005115969083212027539, −2.68131446765903919119403957203, −1.42899517285652978114011759987, −1.40025426974434346855938393620, 0, 0,
1.40025426974434346855938393620, 1.42899517285652978114011759987, 2.68131446765903919119403957203, 2.75399160005115969083212027539, 3.90347600751386956696627429261, 4.06025702827235363212895873806, 4.91358242474775315525359673757, 5.20819866167320386195214415906, 5.69762026080722578143651562477, 5.99730234451416221997723952991, 6.70384028344584700557302357391, 6.82572561928684254092992612251, 7.40761989811732388374900332511, 7.43550692007863997360746483486, 8.779546121922510928690102687875, 8.884763345598069156319389002724, 9.292169009989587146818572595639, 9.333587887996429382294698081494
