| L(s) = 1 | − 3·4-s − 3·9-s + 6·16-s + 8·19-s + 25-s − 28·29-s − 24·31-s + 9·36-s − 12·41-s + 14·49-s − 4·59-s − 28·61-s − 10·64-s + 12·71-s − 24·76-s + 8·79-s + 6·81-s − 16·89-s − 3·100-s + 44·101-s + 12·109-s + 84·116-s − 34·121-s + 72·124-s + 16·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 9-s + 3/2·16-s + 1.83·19-s + 1/5·25-s − 5.19·29-s − 4.31·31-s + 3/2·36-s − 1.87·41-s + 2·49-s − 0.520·59-s − 3.58·61-s − 5/4·64-s + 1.42·71-s − 2.75·76-s + 0.900·79-s + 2/3·81-s − 1.69·89-s − 0.299·100-s + 4.37·101-s + 1.14·109-s + 7.79·116-s − 3.09·121-s + 6.46·124-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5646329139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5646329139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 23 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( 1 - 2 p T^{2} + 127 T^{4} - 1028 T^{6} + 127 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 17 T^{2} - 16 T^{3} + 17 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 46 T^{2} + 1063 T^{4} - 16228 T^{6} + 1063 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 58 T^{2} + 1455 T^{4} - 25708 T^{6} + 1455 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 4 T + 49 T^{2} - 136 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 14 T + 91 T^{2} + 468 T^{3} + 91 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 4 T + p T^{2} )^{6} \) |
| 37 | \( 1 - 18 T^{2} - 265 T^{4} + 21412 T^{6} - 265 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 6 T + 71 T^{2} + 500 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 150 T^{2} + 9335 T^{4} - 407060 T^{6} + 9335 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 90 T^{2} + 5231 T^{4} - 236204 T^{6} + 5231 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 258 T^{2} + 29975 T^{4} - 2025596 T^{6} + 29975 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 2 T + 93 T^{2} + 132 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 14 T + 115 T^{2} + 596 T^{3} + 115 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 198 T^{2} + 22055 T^{4} - 1725428 T^{6} + 22055 p^{2} T^{8} - 198 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 6 T + 89 T^{2} - 1084 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 310 T^{2} + 45631 T^{4} - 4119796 T^{6} + 45631 p^{2} T^{8} - 310 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 4 T + 93 T^{2} - 40 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 322 T^{2} + 51047 T^{4} - 5155260 T^{6} + 51047 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 8 T + 235 T^{2} + 1296 T^{3} + 235 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 330 T^{2} + 57999 T^{4} - 6805708 T^{6} + 57999 p^{2} T^{8} - 330 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.49288341192927043017928742824, −5.44541610131936594297340118509, −5.43706490544826856756977452236, −5.11996696825022873012102947786, −4.94530519209386311746993164876, −4.79602761902937948653697760210, −4.67200353072705918438445387483, −4.42056191506382165798247338144, −4.01822819301975809525106485614, −3.97279167040471620375744686712, −3.80224340879444563616649041099, −3.70957367324750481010698043165, −3.36444271991513187527168419979, −3.32651984691583597331848678487, −3.31249983328288677068321483242, −3.04720196881521653527097991728, −2.71571553421871008147453122111, −2.20330752512858384749545525974, −2.09291973819645089464573489828, −1.99778208585711164553479391031, −1.57560809319713457757397663122, −1.57517351925560974559097061736, −1.10442929055303463146516145853, −0.36468855572919945059992450275, −0.31488607278915811316398700563,
0.31488607278915811316398700563, 0.36468855572919945059992450275, 1.10442929055303463146516145853, 1.57517351925560974559097061736, 1.57560809319713457757397663122, 1.99778208585711164553479391031, 2.09291973819645089464573489828, 2.20330752512858384749545525974, 2.71571553421871008147453122111, 3.04720196881521653527097991728, 3.31249983328288677068321483242, 3.32651984691583597331848678487, 3.36444271991513187527168419979, 3.70957367324750481010698043165, 3.80224340879444563616649041099, 3.97279167040471620375744686712, 4.01822819301975809525106485614, 4.42056191506382165798247338144, 4.67200353072705918438445387483, 4.79602761902937948653697760210, 4.94530519209386311746993164876, 5.11996696825022873012102947786, 5.43706490544826856756977452236, 5.44541610131936594297340118509, 5.49288341192927043017928742824
Plot not available for L-functions of degree greater than 10.
