| L(s) = 1 | − 3·2-s + 6·3-s + 5·4-s + 6·5-s − 18·6-s + 14·7-s − 27·8-s + 27·9-s − 18·10-s − 6·11-s + 30·12-s + 16·13-s − 42·14-s + 36·15-s + 69·16-s − 6·17-s − 81·18-s + 64·19-s + 30·20-s + 84·21-s + 18·22-s + 6·23-s − 162·24-s − 166·25-s − 48·26-s + 108·27-s + 70·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1.15·3-s + 5/8·4-s + 0.536·5-s − 1.22·6-s + 0.755·7-s − 1.19·8-s + 9-s − 0.569·10-s − 0.164·11-s + 0.721·12-s + 0.341·13-s − 0.801·14-s + 0.619·15-s + 1.07·16-s − 0.0856·17-s − 1.06·18-s + 0.772·19-s + 0.335·20-s + 0.872·21-s + 0.174·22-s + 0.0543·23-s − 1.37·24-s − 1.32·25-s − 0.362·26-s + 0.769·27-s + 0.472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.057700512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057700512\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 3 T + p^{2} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 6 T + 202 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 1246 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 16 T + 2406 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 9778 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 64 T + 6534 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 7870 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 252 T + 56446 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 40 T - 13890 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 248 T + 98214 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 450 T + 175642 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 141790 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1104 T + 602230 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 804 T + 380614 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 428 T + 425886 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 148 T + 440790 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 954 T + 13106 p T^{2} - 954 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1072 T + 1063278 T^{2} - 1072 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 572 T + 901662 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1944 T + 1957030 T^{2} - 1944 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 366 T + 1156090 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 808 T + 903054 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−17.98754763658970438917881986449, −17.57422945386022424369784705375, −17.04459096373884350284302281883, −15.98086122027751495503143187930, −15.48248129646803441342299119708, −14.95781607476672701284447506651, −14.11173331179018455937931545626, −13.73575929130420354671972248254, −12.86151962394381771607778953262, −11.95403214843741524545736968147, −11.21421215607790565009657466779, −10.29660072704947938309086752171, −9.382639145802876843343958417977, −9.243114880363181102766207436345, −8.177937000406163814747537789084, −7.77893703428859490642609174030, −6.57897767162756355612751119591, −5.34722196807412904954605676407, −3.52383899701414130274827904738, −1.97603540068635921626971898304,
1.97603540068635921626971898304, 3.52383899701414130274827904738, 5.34722196807412904954605676407, 6.57897767162756355612751119591, 7.77893703428859490642609174030, 8.177937000406163814747537789084, 9.243114880363181102766207436345, 9.382639145802876843343958417977, 10.29660072704947938309086752171, 11.21421215607790565009657466779, 11.95403214843741524545736968147, 12.86151962394381771607778953262, 13.73575929130420354671972248254, 14.11173331179018455937931545626, 14.95781607476672701284447506651, 15.48248129646803441342299119708, 15.98086122027751495503143187930, 17.04459096373884350284302281883, 17.57422945386022424369784705375, 17.98754763658970438917881986449
