| L(s) = 1 | + 4·2-s + 7·4-s + 7·8-s + 9·11-s + 7·16-s + 36·22-s + 11·23-s − 15·25-s + 15·29-s + 14·32-s − 11·37-s − 43-s + 63·44-s + 44·46-s − 60·50-s + 9·53-s + 60·58-s + 28·64-s + 19·67-s + 15·71-s − 44·74-s + 17·79-s − 4·86-s + 63·88-s + 77·92-s − 105·100-s + 36·106-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 7/2·4-s + 2.47·8-s + 2.71·11-s + 7/4·16-s + 7.67·22-s + 2.29·23-s − 3·25-s + 2.78·29-s + 2.47·32-s − 1.80·37-s − 0.152·43-s + 9.49·44-s + 6.48·46-s − 8.48·50-s + 1.23·53-s + 7.87·58-s + 7/2·64-s + 2.32·67-s + 1.78·71-s − 5.11·74-s + 1.91·79-s − 0.431·86-s + 6.71·88-s + 8.02·92-s − 10.5·100-s + 3.49·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(29.43491366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.43491366\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_6$ | \( 1 - p^{2} T + 9 T^{2} - 15 T^{3} + 9 p T^{4} - p^{4} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 11 | $C_6$ | \( 1 - 9 T + 53 T^{2} - 211 T^{3} + 53 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 23 | $C_6$ | \( 1 - 11 T + 93 T^{2} - 477 T^{3} + 93 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_6$ | \( 1 - 15 T + 113 T^{2} - 659 T^{3} + 113 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 37 | $C_6$ | \( 1 + 11 T + 9 T^{2} - 265 T^{3} + 9 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 43 | $C_6$ | \( 1 + T - 27 T^{2} - 293 T^{3} - 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 53 | $C_6$ | \( 1 - 9 T - 31 T^{2} + 643 T^{3} - 31 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 67 | $C_6$ | \( 1 - 19 T + 109 T^{2} - 265 T^{3} + 109 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_6$ | \( 1 - 15 T + 197 T^{2} - 1513 T^{3} + 197 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 79 | $C_6$ | \( 1 - 17 T + 37 T^{2} + 673 T^{3} + 37 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50955877166078198185030267690, −7.22842578248106846204210845099, −6.94250687838718004469102453218, −6.87094736714219610876679045354, −6.42524603037223200652341465293, −6.38015201294494446472187689750, −6.25657783361194362387890692977, −5.68546649189159598296953744818, −5.47149361047770508280314573286, −5.45439346302643476216164654637, −4.89481571053025270147292457082, −4.80932642420606617790971185649, −4.67030453927734455630607833899, −4.08131917121667584409392373751, −4.04712632397010636545020001038, −3.92908336003531947022511649019, −3.42250778983192700099441856697, −3.32264246125395514731382553303, −3.25324289369571635484143627577, −2.41086694049968221534765454802, −2.39747507476551997274644725474, −1.82364688034483973152107590677, −1.47531795364083772793048503215, −0.875861957043457171824047561979, −0.78694206905802527860061476517,
0.78694206905802527860061476517, 0.875861957043457171824047561979, 1.47531795364083772793048503215, 1.82364688034483973152107590677, 2.39747507476551997274644725474, 2.41086694049968221534765454802, 3.25324289369571635484143627577, 3.32264246125395514731382553303, 3.42250778983192700099441856697, 3.92908336003531947022511649019, 4.04712632397010636545020001038, 4.08131917121667584409392373751, 4.67030453927734455630607833899, 4.80932642420606617790971185649, 4.89481571053025270147292457082, 5.45439346302643476216164654637, 5.47149361047770508280314573286, 5.68546649189159598296953744818, 6.25657783361194362387890692977, 6.38015201294494446472187689750, 6.42524603037223200652341465293, 6.87094736714219610876679045354, 6.94250687838718004469102453218, 7.22842578248106846204210845099, 7.50955877166078198185030267690
