L(s) = 1 | + 2·2-s − 2·3-s − 5·4-s − 4·6-s − 12·8-s + 9-s + 10·12-s + 15·16-s + 2·18-s + 24·24-s − 2·25-s + 2·31-s + 42·32-s − 5·36-s − 2·41-s + 2·47-s − 30·48-s − 2·49-s − 4·50-s + 4·62-s − 35·64-s − 12·72-s + 2·73-s + 4·75-s − 4·82-s − 4·93-s + 4·94-s + ⋯ |
L(s) = 1 | + 2·2-s − 2·3-s − 5·4-s − 4·6-s − 12·8-s + 9-s + 10·12-s + 15·16-s + 2·18-s + 24·24-s − 2·25-s + 2·31-s + 42·32-s − 5·36-s − 2·41-s + 2·47-s − 30·48-s − 2·49-s − 4·50-s + 4·62-s − 35·64-s − 12·72-s + 2·73-s + 4·75-s − 4·82-s − 4·93-s + 4·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1290584732\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1290584732\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 23 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 29 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
good | 2 | \( ( 1 + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 11 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \) |
| 13 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{6} \) |
| 19 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \) |
| 37 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \) |
| 41 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \) |
| 43 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \) |
| 53 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 61 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \) |
| 79 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \) |
| 83 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 89 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \) |
| 97 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.25909593652186449835958884096, −3.18656055333057495945095168921, −2.97453377920416241530484285525, −2.90724172506648742573410154711, −2.86376255901029625772121901024, −2.77208973828410978020221550142, −2.69325925309115995567696262578, −2.54032805458544618333110414543, −2.33713761154925118500050241129, −2.27338083380008952489515309338, −2.09124909154696315457701405354, −2.07233789720136915399957026876, −2.04307029592682950993425709034, −1.80996401662873055210617973241, −1.70772850594626610419998659872, −1.48041989502667604295017630805, −1.33062390745457513506420992777, −1.27214741778403983367917596258, −1.20406104806879388650302671359, −1.17947667341555740482404024951, −1.16694647723800286705243893062, −0.72271285857536692083285843453, −0.47403144761599708117908831176, −0.46996291595628832652183201979, −0.40056080148540021042572560132,
0.40056080148540021042572560132, 0.46996291595628832652183201979, 0.47403144761599708117908831176, 0.72271285857536692083285843453, 1.16694647723800286705243893062, 1.17947667341555740482404024951, 1.20406104806879388650302671359, 1.27214741778403983367917596258, 1.33062390745457513506420992777, 1.48041989502667604295017630805, 1.70772850594626610419998659872, 1.80996401662873055210617973241, 2.04307029592682950993425709034, 2.07233789720136915399957026876, 2.09124909154696315457701405354, 2.27338083380008952489515309338, 2.33713761154925118500050241129, 2.54032805458544618333110414543, 2.69325925309115995567696262578, 2.77208973828410978020221550142, 2.86376255901029625772121901024, 2.90724172506648742573410154711, 2.97453377920416241530484285525, 3.18656055333057495945095168921, 3.25909593652186449835958884096
Plot not available for L-functions of degree greater than 10.