| L(s) = 1 | − 42·7-s − 27·9-s − 16·17-s + 60·23-s + 306·25-s + 552·31-s − 272·41-s + 1.57e3·47-s + 1.02e3·49-s + 1.13e3·63-s + 2.54e3·71-s + 444·73-s + 3.52e3·79-s + 486·81-s + 16·89-s + 1.54e3·97-s + 2.25e3·103-s + 2.15e3·113-s + 672·119-s + 1.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 432·153-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 9-s − 0.228·17-s + 0.543·23-s + 2.44·25-s + 3.19·31-s − 1.03·41-s + 4.89·47-s + 3·49-s + 2.26·63-s + 4.25·71-s + 0.711·73-s + 5.02·79-s + 2/3·81-s + 0.0190·89-s + 1.62·97-s + 2.15·103-s + 1.79·113-s + 0.517·119-s + 1.19·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.228·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(27.46985298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.46985298\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| 7 | \( ( 1 + p T )^{6} \) |
| good | 5 | \( 1 - 306 T^{2} + 69723 T^{4} - 9689364 T^{6} + 69723 p^{6} T^{8} - 306 p^{12} T^{10} + p^{18} T^{12} \) |
| 11 | \( 1 - 1590 T^{2} - 1291749 T^{4} + 5263277892 T^{6} - 1291749 p^{6} T^{8} - 1590 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 - 8154 T^{2} + 34676871 T^{4} - 94493698924 T^{6} + 34676871 p^{6} T^{8} - 8154 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( ( 1 + 8 T + 11693 T^{2} + 12516 T^{3} + 11693 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( 1 - 22698 T^{2} + 281976327 T^{4} - 6387643244 p^{2} T^{6} + 281976327 p^{6} T^{8} - 22698 p^{12} T^{10} + p^{18} T^{12} \) |
| 23 | \( ( 1 - 30 T + 15315 T^{2} - 1200468 T^{3} + 15315 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 29 | \( 1 - 139878 T^{2} + 8294180007 T^{4} - 267182314202772 T^{6} + 8294180007 p^{6} T^{8} - 139878 p^{12} T^{10} + p^{18} T^{12} \) |
| 31 | \( ( 1 - 276 T + 92193 T^{2} - 15530744 T^{3} + 92193 p^{3} T^{4} - 276 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 294102 T^{2} + 36507613047 T^{4} - 2449192702016884 T^{6} + 36507613047 p^{6} T^{8} - 294102 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 136 T + 73301 T^{2} + 9195708 T^{3} + 73301 p^{3} T^{4} + 136 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 402630 T^{2} + 72488318535 T^{4} - 7436672769249652 T^{6} + 72488318535 p^{6} T^{8} - 402630 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( ( 1 - 788 T + 504413 T^{2} - 177469464 T^{3} + 504413 p^{3} T^{4} - 788 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 53 | \( 1 - 301542 T^{2} + 68147631831 T^{4} - 12599714107370068 T^{6} + 68147631831 p^{6} T^{8} - 301542 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( 1 - 972930 T^{2} + 421846234695 T^{4} - 108520578062446524 T^{6} + 421846234695 p^{6} T^{8} - 972930 p^{12} T^{10} + p^{18} T^{12} \) |
| 61 | \( 1 - 1117626 T^{2} + 551555340327 T^{4} - 158612169634895020 T^{6} + 551555340327 p^{6} T^{8} - 1117626 p^{12} T^{10} + p^{18} T^{12} \) |
| 67 | \( 1 - 997062 T^{2} + 589838394711 T^{4} - 213127652236255540 T^{6} + 589838394711 p^{6} T^{8} - 997062 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 1274 T + 1393979 T^{2} - 907718700 T^{3} + 1393979 p^{3} T^{4} - 1274 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 - 222 T + 504927 T^{2} + 53720844 T^{3} + 504927 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( ( 1 - 1764 T + 2447493 T^{2} - 24014536 p T^{3} + 2447493 p^{3} T^{4} - 1764 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 2959458 T^{2} + 3870450759639 T^{4} - 2863930488145656892 T^{6} + 3870450759639 p^{6} T^{8} - 2959458 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 - 8 T + 1048133 T^{2} - 10589436 T^{3} + 1048133 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( ( 1 - 774 T + 2031927 T^{2} - 1519826180 T^{3} + 2031927 p^{3} T^{4} - 774 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−4.95636044691119473861773287425, −4.32248418486334951042440579109, −4.26032224424791044973091133288, −4.11438901302923093839767796475, −3.95865436252450690146902397095, −3.91939198688195673496808511404, −3.67364173845597576739704860297, −3.42978875653429663505599539866, −3.31932157154225688693528894149, −3.06200904632881474732560586906, −2.90166834669383786260111222801, −2.89142328240186905875591752782, −2.78916610449068454943706361633, −2.49833661136670240110265349502, −2.42829458231761548103115243773, −2.02833668579081431493455182242, −2.02710720029623164806076714069, −1.67986222372957237395272185244, −1.65321391496088608144367557866, −0.803153507669558084594089956797, −0.70596965120411915090365517412, −0.69726987852685125227083853783, −0.68478191193433452821901419265, −0.68223341031286990427632238292, −0.44459311519097580312742998148,
0.44459311519097580312742998148, 0.68223341031286990427632238292, 0.68478191193433452821901419265, 0.69726987852685125227083853783, 0.70596965120411915090365517412, 0.803153507669558084594089956797, 1.65321391496088608144367557866, 1.67986222372957237395272185244, 2.02710720029623164806076714069, 2.02833668579081431493455182242, 2.42829458231761548103115243773, 2.49833661136670240110265349502, 2.78916610449068454943706361633, 2.89142328240186905875591752782, 2.90166834669383786260111222801, 3.06200904632881474732560586906, 3.31932157154225688693528894149, 3.42978875653429663505599539866, 3.67364173845597576739704860297, 3.91939198688195673496808511404, 3.95865436252450690146902397095, 4.11438901302923093839767796475, 4.26032224424791044973091133288, 4.32248418486334951042440579109, 4.95636044691119473861773287425
Plot not available for L-functions of degree greater than 10.
