| L(s) = 1 | + 6·2-s + 21·4-s + 56·8-s − 4·9-s + 8·11-s + 126·16-s − 24·18-s + 48·22-s − 8·23-s − 10·25-s − 6·29-s + 252·32-s − 84·36-s − 12·37-s + 12·43-s + 168·44-s − 48·46-s − 60·50-s + 16·53-s − 36·58-s + 462·64-s − 8·67-s + 48·71-s − 224·72-s − 72·74-s + 64·79-s + 72·86-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 21/2·4-s + 19.7·8-s − 4/3·9-s + 2.41·11-s + 63/2·16-s − 5.65·18-s + 10.2·22-s − 1.66·23-s − 2·25-s − 1.11·29-s + 44.5·32-s − 14·36-s − 1.97·37-s + 1.82·43-s + 25.3·44-s − 7.07·46-s − 8.48·50-s + 2.19·53-s − 4.72·58-s + 57.7·64-s − 0.977·67-s + 5.69·71-s − 26.3·72-s − 8.36·74-s + 7.20·79-s + 7.76·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(192.0993231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(192.0993231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{6} \) |
| 7 | \( 1 \) |
| 29 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 + 4 T^{2} + 16 T^{4} + 20 p T^{6} + 16 p^{2} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 2 p T^{2} + 72 T^{4} + 342 T^{6} + 72 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 4 T + 26 T^{2} - 70 T^{3} + 26 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 18 T^{2} + 40 p T^{4} + 5998 T^{6} + 40 p^{3} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 52 T^{2} + 1575 T^{4} + 32808 T^{6} + 1575 p^{2} T^{8} + 52 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 52 T^{2} + 1787 T^{4} + 39080 T^{6} + 1787 p^{2} T^{8} + 52 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 4 T + 25 T^{2} + 152 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 118 T^{2} + 6896 T^{4} + 257746 T^{6} + 6896 p^{2} T^{8} + 118 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 41 | \( 1 + 116 T^{2} + 6327 T^{4} + 266152 T^{6} + 6327 p^{2} T^{8} + 116 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 6 T - 14 T^{2} + 442 T^{3} - 14 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 214 T^{2} + 21264 T^{4} + 1258418 T^{6} + 21264 p^{2} T^{8} + 214 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 8 T + 168 T^{2} - 838 T^{3} + 168 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 128 T^{2} + 14847 T^{4} + 912016 T^{6} + 14847 p^{2} T^{8} + 128 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 10 T^{2} + 10935 T^{4} - 74060 T^{6} + 10935 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 4 T + 141 T^{2} + 392 T^{3} + 141 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 8 T + p T^{2} )^{6} \) |
| 73 | \( 1 + 212 T^{2} + 29911 T^{4} + 2525992 T^{6} + 29911 p^{2} T^{8} + 212 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 32 T + 562 T^{2} - 6088 T^{3} + 562 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 368 T^{2} + 62607 T^{4} + 6465520 T^{6} + 62607 p^{2} T^{8} + 368 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 388 T^{2} + 70727 T^{4} + 7850056 T^{6} + 70727 p^{2} T^{8} + 388 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 196 T^{2} - 2137 T^{4} - 2280760 T^{6} - 2137 p^{2} T^{8} + 196 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.68725723757493490708764875030, −4.18446722328797631862816600855, −4.05235642476626868692892516767, −4.03937904554846671055786585767, −3.93378573157589509064851509659, −3.89343804157521814230061146116, −3.82118944347297926172892815682, −3.67126751878438767499994749327, −3.43827945716487925257140043107, −3.40806610578006005827783985428, −3.14943056707943255177058037145, −3.05172922911539021457222695358, −2.97307187068596459852382407160, −2.41694688383883644122127003916, −2.28453215875001192344323639235, −2.25519050548202502953563441813, −2.21297219260192775001974318205, −2.21116496827409274494092921990, −1.91846924207151736668267198554, −1.60516856580702601632244991063, −1.50911126064617855427723972297, −1.22157285248571859238477241639, −0.813177630844367834486084504674, −0.71570371196729240742024359716, −0.37768379326573348041855882458,
0.37768379326573348041855882458, 0.71570371196729240742024359716, 0.813177630844367834486084504674, 1.22157285248571859238477241639, 1.50911126064617855427723972297, 1.60516856580702601632244991063, 1.91846924207151736668267198554, 2.21116496827409274494092921990, 2.21297219260192775001974318205, 2.25519050548202502953563441813, 2.28453215875001192344323639235, 2.41694688383883644122127003916, 2.97307187068596459852382407160, 3.05172922911539021457222695358, 3.14943056707943255177058037145, 3.40806610578006005827783985428, 3.43827945716487925257140043107, 3.67126751878438767499994749327, 3.82118944347297926172892815682, 3.89343804157521814230061146116, 3.93378573157589509064851509659, 4.03937904554846671055786585767, 4.05235642476626868692892516767, 4.18446722328797631862816600855, 4.68725723757493490708764875030
Plot not available for L-functions of degree greater than 10.
