L(s) = 1 | + 2·2-s + 8·3-s + 4-s + 5-s + 16·6-s − 2·7-s + 28·9-s + 2·10-s + 8·12-s − 4·14-s + 8·15-s + 56·18-s + 20-s − 16·21-s + 5·23-s + 25-s + 50·27-s − 2·28-s − 29-s + 16·30-s − 2·35-s + 28·36-s + 2·41-s − 32·42-s − 43-s + 28·45-s + 10·46-s + ⋯ |
L(s) = 1 | + 2·2-s + 8·3-s + 4-s + 5-s + 16·6-s − 2·7-s + 28·9-s + 2·10-s + 8·12-s − 4·14-s + 8·15-s + 56·18-s + 20-s − 16·21-s + 5·23-s + 25-s + 50·27-s − 2·28-s − 29-s + 16·30-s − 2·35-s + 28·36-s + 2·41-s − 32·42-s − 43-s + 28·45-s + 10·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(14.58847319\) |
\(L(\frac12)\) |
\(\approx\) |
\(14.58847319\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 5 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| 43 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
good | 3 | \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 7 | \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 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 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 23 | \( ( 1 - T + T^{2} )^{6}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 31 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 37 | \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 47 | \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 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 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 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 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 79 | \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \) |
| 83 | \( ( 1 + T )^{12}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 89 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 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
−3.55255276097083559006401946903, −3.27661094886418478595446179783, −3.25241427279881856296926005769, −3.18389386223429966303289185339, −3.11214082546378941010459256042, −3.05073408645484942892364936556, −2.95053386850229251468559918482, −2.88272003894386209289908642274, −2.86253089994093107075604340291, −2.73749364301267328143968306762, −2.66682916674608402688546306649, −2.66182304784523456561600305591, −2.59723643374018159068876439452, −2.45456965404157718072420768018, −2.33886258303980848592740659706, −2.24866252555050726512828342552, −2.08609819249974116292272852191, −1.99187429852747176961725041964, −1.77756017022405434991753819640, −1.71376555259076350538037276390, −1.57208906255589500114868817904, −1.53111046392020609127624822629, −1.17269331892306460521559677414, −1.14897552008556557074794345928, −1.01247585237578418648276404020,
1.01247585237578418648276404020, 1.14897552008556557074794345928, 1.17269331892306460521559677414, 1.53111046392020609127624822629, 1.57208906255589500114868817904, 1.71376555259076350538037276390, 1.77756017022405434991753819640, 1.99187429852747176961725041964, 2.08609819249974116292272852191, 2.24866252555050726512828342552, 2.33886258303980848592740659706, 2.45456965404157718072420768018, 2.59723643374018159068876439452, 2.66182304784523456561600305591, 2.66682916674608402688546306649, 2.73749364301267328143968306762, 2.86253089994093107075604340291, 2.88272003894386209289908642274, 2.95053386850229251468559918482, 3.05073408645484942892364936556, 3.11214082546378941010459256042, 3.18389386223429966303289185339, 3.25241427279881856296926005769, 3.27661094886418478595446179783, 3.55255276097083559006401946903
Plot not available for L-functions of degree greater than 10.