| L(s) = 1 | + 2·3-s − 3·4-s + 9-s − 6·12-s + 6·16-s − 2·23-s − 3·25-s − 2·29-s − 3·36-s − 2·43-s + 12·48-s + 49-s − 2·61-s − 10·64-s − 4·69-s − 6·75-s − 4·87-s + 6·92-s + 9·100-s + 2·101-s − 2·103-s − 12·107-s + 6·116-s − 6·121-s + 127-s − 4·129-s + 131-s + ⋯ |
| L(s) = 1 | + 2·3-s − 3·4-s + 9-s − 6·12-s + 6·16-s − 2·23-s − 3·25-s − 2·29-s − 3·36-s − 2·43-s + 12·48-s + 49-s − 2·61-s − 10·64-s − 4·69-s − 6·75-s − 4·87-s + 6·92-s + 9·100-s + 2·101-s − 2·103-s − 12·107-s + 6·116-s − 6·121-s + 127-s − 4·129-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2668850907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2668850907\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 \) |
| good | 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 7 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 11 | \( ( 1 + T^{2} )^{6} \) |
| 17 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 19 | \( ( 1 + T^{2} )^{6} \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 31 | \( ( 1 + T^{2} )^{6} \) |
| 37 | \( ( 1 + T^{2} )^{6} \) |
| 41 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 43 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 47 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 53 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 59 | \( ( 1 + T^{2} )^{6} \) |
| 61 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 67 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 71 | \( ( 1 + T^{2} )^{6} \) |
| 73 | \( ( 1 + T^{2} )^{6} \) |
| 79 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 83 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 89 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 97 | \( ( 1 + T^{2} )^{6} \) |
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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
−4.35430304724716771314273027388, −4.31826993648652390912261914819, −4.26738629385921988574379984493, −4.19360945102866470251702266727, −4.06406645730127462301452459236, −3.94532162020417473331221551151, −3.75073924929334645425149785715, −3.74662933276159168977909475357, −3.57816615695919598562017957795, −3.42496891233539328951198816455, −3.28695519374942762769133435910, −2.99553382396427557280200997703, −2.85899966171900901489236454785, −2.80648897470402476188887719907, −2.47902888687541721978926717957, −2.45522742819875414823543959313, −2.42829251813710405773995315413, −1.98884143501409551172753126235, −1.78106214904955480164483772000, −1.46998425567382240049563333863, −1.45785148237982570896569069161, −1.32888475491935003320179518396, −1.32077722154743336168691680400, −0.36076550608344183664642903448, −0.27346358729708911816249016841,
0.27346358729708911816249016841, 0.36076550608344183664642903448, 1.32077722154743336168691680400, 1.32888475491935003320179518396, 1.45785148237982570896569069161, 1.46998425567382240049563333863, 1.78106214904955480164483772000, 1.98884143501409551172753126235, 2.42829251813710405773995315413, 2.45522742819875414823543959313, 2.47902888687541721978926717957, 2.80648897470402476188887719907, 2.85899966171900901489236454785, 2.99553382396427557280200997703, 3.28695519374942762769133435910, 3.42496891233539328951198816455, 3.57816615695919598562017957795, 3.74662933276159168977909475357, 3.75073924929334645425149785715, 3.94532162020417473331221551151, 4.06406645730127462301452459236, 4.19360945102866470251702266727, 4.26738629385921988574379984493, 4.31826993648652390912261914819, 4.35430304724716771314273027388
Plot not available for L-functions of degree greater than 10.
