| L(s) = 1 | + 6·3-s + 6·5-s + 10·7-s + 27·9-s − 22·11-s + 2·13-s + 36·15-s − 104·17-s − 4·19-s + 60·21-s − 102·23-s − 206·25-s + 108·27-s + 392·29-s − 64·31-s − 132·33-s + 60·35-s + 164·37-s + 12·39-s − 732·41-s − 168·43-s + 162·45-s − 314·47-s − 594·49-s − 624·51-s + 382·53-s − 132·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.536·5-s + 0.539·7-s + 9-s − 0.603·11-s + 0.0426·13-s + 0.619·15-s − 1.48·17-s − 0.0482·19-s + 0.623·21-s − 0.924·23-s − 1.64·25-s + 0.769·27-s + 2.51·29-s − 0.370·31-s − 0.696·33-s + 0.289·35-s + 0.728·37-s + 0.0492·39-s − 2.78·41-s − 0.595·43-s + 0.536·45-s − 0.974·47-s − 1.73·49-s − 1.71·51-s + 0.990·53-s − 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T + 242 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 10 T + 694 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 518 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 104 T + 7022 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T - 1578 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 102 T + 24062 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 392 T + 75702 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 64 T + 33406 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 164 T - 770 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 732 T + 264998 T^{2} + 732 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 168 T + 2034 p T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 314 T + 4210 p T^{2} + 314 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 382 T + 310962 T^{2} - 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 508 T + 418086 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 66354 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 216 T + 298758 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 878 T + 841070 T^{2} + 878 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 260 T - 58058 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 118 T + 829606 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 496 T + 441030 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1756 T + 1701014 T^{2} + 1756 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1968 T + 2769054 T^{2} - 1968 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.411767325742697806933280181462, −8.337489322325471838184407047708, −7.82974569617000142788824232261, −7.63677027791231459218617039427, −6.90668145349513439294869556249, −6.70489132155268312508189300355, −6.14130110728928198635599652796, −6.01802430917160566364053382828, −5.10823079192003126314106126998, −4.89900040533400795602638769789, −4.54218446818073116428382117056, −4.03199578112362212484664935289, −3.49822120349542492535499791999, −3.08752555256402779561180415397, −2.38038214960761178044268491074, −2.30335863360138121773310135049, −1.50810991038328436082166500723, −1.42170646097473393085516616965, 0, 0,
1.42170646097473393085516616965, 1.50810991038328436082166500723, 2.30335863360138121773310135049, 2.38038214960761178044268491074, 3.08752555256402779561180415397, 3.49822120349542492535499791999, 4.03199578112362212484664935289, 4.54218446818073116428382117056, 4.89900040533400795602638769789, 5.10823079192003126314106126998, 6.01802430917160566364053382828, 6.14130110728928198635599652796, 6.70489132155268312508189300355, 6.90668145349513439294869556249, 7.63677027791231459218617039427, 7.82974569617000142788824232261, 8.337489322325471838184407047708, 8.411767325742697806933280181462
