| L(s) = 1 | + 3-s − 7-s + 9-s + 19-s − 21-s − 25-s − 31-s + 8·37-s + 49-s + 57-s + 7·61-s − 63-s − 3·67-s + 3·73-s − 75-s + 3·79-s − 93-s − 103-s − 109-s + 8·111-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 147-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s + 9-s + 19-s − 21-s − 25-s − 31-s + 8·37-s + 49-s + 57-s + 7·61-s − 63-s − 3·67-s + 3·73-s − 75-s + 3·79-s − 93-s − 103-s − 109-s + 8·111-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.708500768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708500768\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| 7 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| good | 5 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 11 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 17 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 23 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 31 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 37 | \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 43 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 59 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 61 | \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 67 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 89 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
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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
−2.93986864003179983734631360634, −2.93356985687919899681067601727, −2.81129816185959052467601574798, −2.67368809637358748314230776404, −2.53848156751059836962600290622, −2.50973342558212575207553800728, −2.44628118836748283791890912465, −2.40618838569707255347993940867, −2.30123226132333570460580317897, −2.25797877292481663635480540608, −2.21129774804057399262417619511, −2.08049248179818631816501538390, −2.06154610558065576823859264431, −2.05913513792703603061356779771, −1.75809069991171989920483295506, −1.50637813220103621201778098390, −1.33353574528254700881290457374, −1.31357817656027967580070081997, −1.25300934939407960684611742031, −1.14468434830964565516434189602, −1.13160502629059009106341582093, −0.939974867931265672778694289678, −0.808216166066214732928758217788, −0.73124525254965483192567757554, −0.37826742947040334565485987556,
0.37826742947040334565485987556, 0.73124525254965483192567757554, 0.808216166066214732928758217788, 0.939974867931265672778694289678, 1.13160502629059009106341582093, 1.14468434830964565516434189602, 1.25300934939407960684611742031, 1.31357817656027967580070081997, 1.33353574528254700881290457374, 1.50637813220103621201778098390, 1.75809069991171989920483295506, 2.05913513792703603061356779771, 2.06154610558065576823859264431, 2.08049248179818631816501538390, 2.21129774804057399262417619511, 2.25797877292481663635480540608, 2.30123226132333570460580317897, 2.40618838569707255347993940867, 2.44628118836748283791890912465, 2.50973342558212575207553800728, 2.53848156751059836962600290622, 2.67368809637358748314230776404, 2.81129816185959052467601574798, 2.93356985687919899681067601727, 2.93986864003179983734631360634
Plot not available for L-functions of degree greater than 10.
