| L(s) = 1 | − 3·2-s − 2·3-s + 6·4-s + 6·6-s − 10·8-s − 9-s − 12·12-s + 8·13-s + 15·16-s + 3·18-s + 8·19-s + 2·23-s + 20·24-s − 24·26-s + 4·27-s − 2·29-s + 3·31-s − 21·32-s − 6·36-s + 8·37-s − 24·38-s − 16·39-s − 2·41-s + 10·43-s − 6·46-s − 20·47-s − 30·48-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 3·4-s + 2.44·6-s − 3.53·8-s − 1/3·9-s − 3.46·12-s + 2.21·13-s + 15/4·16-s + 0.707·18-s + 1.83·19-s + 0.417·23-s + 4.08·24-s − 4.70·26-s + 0.769·27-s − 0.371·29-s + 0.538·31-s − 3.71·32-s − 36-s + 1.31·37-s − 3.89·38-s − 2.56·39-s − 0.312·41-s + 1.52·43-s − 0.884·46-s − 2.91·47-s − 4.33·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7972258135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7972258135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 52 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 55 T^{2} - 212 T^{3} + 55 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 57 T^{2} - 84 T^{3} + 57 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T - 9 T^{2} - 144 T^{3} - 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 87 T^{2} - 500 T^{3} + 87 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 39 T^{2} + 396 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 69 T^{2} - 256 T^{3} + 69 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 20 T + 221 T^{2} + 1672 T^{3} + 221 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 20 T + 247 T^{2} - 2124 T^{3} + 247 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 20 T + 289 T^{2} - 2520 T^{3} + 289 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 18 T + 263 T^{2} + 2296 T^{3} + 263 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 89 T^{2} - 424 T^{3} + 89 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 181 T^{2} - 1008 T^{3} + 181 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 20 T + 259 T^{2} - 2456 T^{3} + 259 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 45 T^{2} - 160 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 237 T^{2} + 1536 T^{3} + 237 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 311 T^{2} - 3164 T^{3} + 311 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 263 T^{2} - 1932 T^{3} + 263 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.461975890368965836257772309029, −8.170846402149702056110442765312, −7.927641895168107130305696649848, −7.84331671187051968851925496229, −7.23528294367725444693617945610, −7.12605104711710632060938226030, −6.83093326369306850933507587700, −6.44287210493042961057123285503, −6.32860439860796823705357004899, −6.06454278462203014935722696142, −5.64791243747540306576122953779, −5.51040322723956297007352218821, −5.34861737600717971857536925768, −4.89163431214326498525954819598, −4.42488910823582274801202876017, −3.99357868667688471501889835748, −3.48427252930204279519263680205, −3.34073594012998037603360939658, −3.13374077409243595596089969051, −2.37937454656266499056271770888, −2.28362919230070873613025752997, −1.64790297336344649394157629805, −1.06754218158707787195306378599, −0.917491871808395429727144960725, −0.51446926149817869653789047284,
0.51446926149817869653789047284, 0.917491871808395429727144960725, 1.06754218158707787195306378599, 1.64790297336344649394157629805, 2.28362919230070873613025752997, 2.37937454656266499056271770888, 3.13374077409243595596089969051, 3.34073594012998037603360939658, 3.48427252930204279519263680205, 3.99357868667688471501889835748, 4.42488910823582274801202876017, 4.89163431214326498525954819598, 5.34861737600717971857536925768, 5.51040322723956297007352218821, 5.64791243747540306576122953779, 6.06454278462203014935722696142, 6.32860439860796823705357004899, 6.44287210493042961057123285503, 6.83093326369306850933507587700, 7.12605104711710632060938226030, 7.23528294367725444693617945610, 7.84331671187051968851925496229, 7.927641895168107130305696649848, 8.170846402149702056110442765312, 8.461975890368965836257772309029
