| L(s) = 1 | + 6·3-s − 3·4-s + 6·5-s + 15·9-s − 18·12-s + 36·15-s + 6·16-s − 18·20-s + 12·23-s + 21·25-s + 22·27-s − 45·36-s + 90·45-s + 42·47-s + 36·48-s − 33·49-s + 24·53-s − 108·60-s − 10·64-s + 30·67-s + 72·69-s + 126·75-s + 36·80-s + 27·81-s − 30·89-s − 36·92-s + 24·97-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 3/2·4-s + 2.68·5-s + 5·9-s − 5.19·12-s + 9.29·15-s + 3/2·16-s − 4.02·20-s + 2.50·23-s + 21/5·25-s + 4.23·27-s − 7.5·36-s + 13.4·45-s + 6.12·47-s + 5.19·48-s − 4.71·49-s + 3.29·53-s − 13.9·60-s − 5/4·64-s + 3.66·67-s + 8.66·69-s + 14.5·75-s + 4.02·80-s + 3·81-s − 3.17·89-s − 3.75·92-s + 2.43·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(24.56456712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.56456712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 - T )^{6} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3 T^{2} + 3 T^{4} + T^{6} + 3 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( ( 1 - p T + 2 p T^{2} - 11 T^{3} + 2 p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 + 33 T^{2} + 498 T^{4} + 4421 T^{6} + 498 p^{2} T^{8} + 33 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 + 6 T^{2} + 87 T^{4} + 740 T^{6} + 87 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 54 T^{2} + 1455 T^{4} + 27316 T^{6} + 1455 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 78 T^{2} + 2775 T^{4} + 62804 T^{6} + 2775 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 6 T + 57 T^{2} - 192 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 + 30 T^{2} + 1095 T^{4} + 54916 T^{6} + 1095 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 45 T^{2} - 4 p T^{3} + 45 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 87 T^{2} - 16 T^{3} + 87 p T^{4} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 141 T^{2} + 10494 T^{4} + 517105 T^{6} + 10494 p^{2} T^{8} + 141 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 + 3 p T^{2} + 10242 T^{4} + 518861 T^{6} + 10242 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 21 T + 6 p T^{2} - 2277 T^{3} + 6 p^{2} T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 12 T + 75 T^{2} - 540 T^{3} + 75 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( ( 1 + 165 T^{2} + 12 T^{3} + 165 p T^{4} + p^{3} T^{6} )^{2} \) |
| 61 | \( 1 + 333 T^{2} + 48102 T^{4} + 3843137 T^{6} + 48102 p^{2} T^{8} + 333 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 15 T + 270 T^{2} - 2107 T^{3} + 270 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 201 T^{2} - 12 T^{3} + 201 p T^{4} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 246 T^{2} + 30015 T^{4} + 2489204 T^{6} + 30015 p^{2} T^{8} + 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 + 198 T^{2} + 17583 T^{4} + 1191188 T^{6} + 17583 p^{2} T^{8} + 198 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 + 186 T^{2} + 14391 T^{4} + 929404 T^{6} + 14391 p^{2} T^{8} + 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 15 T + 246 T^{2} + 2523 T^{3} + 246 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 12 T + 255 T^{2} - 2252 T^{3} + 255 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.83533073575909839658303084099, −5.46390942896371785159299389916, −5.31897558442518255567049197708, −5.11232666843560596293019663467, −5.05274589018104801506303290634, −4.86854612007415761066388000758, −4.74390794043512005718405669203, −4.46280926403351647467884141200, −4.21186716905427695676485631555, −3.93066057255287066774402928073, −3.89556705530935197087215869269, −3.64954194556746814741035459736, −3.49967526285999022435065296196, −3.20079170385645795060093962102, −3.04012580204745052381749147407, −2.88776139173192039341460422885, −2.67245703990020557550375306389, −2.44299394354336474194795575695, −2.37840793565645615555423905524, −2.24498261502305108827566285417, −1.98505365307232056688200426516, −1.53956360492630326633157608259, −1.25138706605810035988356508603, −0.924614641754491117924675868799, −0.794318638946469023667388282742,
0.794318638946469023667388282742, 0.924614641754491117924675868799, 1.25138706605810035988356508603, 1.53956360492630326633157608259, 1.98505365307232056688200426516, 2.24498261502305108827566285417, 2.37840793565645615555423905524, 2.44299394354336474194795575695, 2.67245703990020557550375306389, 2.88776139173192039341460422885, 3.04012580204745052381749147407, 3.20079170385645795060093962102, 3.49967526285999022435065296196, 3.64954194556746814741035459736, 3.89556705530935197087215869269, 3.93066057255287066774402928073, 4.21186716905427695676485631555, 4.46280926403351647467884141200, 4.74390794043512005718405669203, 4.86854612007415761066388000758, 5.05274589018104801506303290634, 5.11232666843560596293019663467, 5.31897558442518255567049197708, 5.46390942896371785159299389916, 5.83533073575909839658303084099
Plot not available for L-functions of degree greater than 10.
