| L(s) = 1 | − 3·4-s − 6·5-s − 3·9-s + 2·11-s + 6·16-s − 2·19-s + 18·20-s + 16·25-s − 12·29-s + 6·31-s + 9·36-s + 12·41-s − 6·44-s + 18·45-s + 9·49-s − 12·55-s − 4·59-s + 32·61-s − 10·64-s + 18·71-s + 6·76-s − 22·79-s − 36·80-s + 6·81-s − 58·89-s + 12·95-s − 6·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 2.68·5-s − 9-s + 0.603·11-s + 3/2·16-s − 0.458·19-s + 4.02·20-s + 16/5·25-s − 2.22·29-s + 1.07·31-s + 3/2·36-s + 1.87·41-s − 0.904·44-s + 2.68·45-s + 9/7·49-s − 1.61·55-s − 0.520·59-s + 4.09·61-s − 5/4·64-s + 2.13·71-s + 0.688·76-s − 2.47·79-s − 4.02·80-s + 2/3·81-s − 6.14·89-s + 1.23·95-s − 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04669767132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04669767132\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + 4 p T^{2} + 2 p^{2} T^{3} + 4 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | \( ( 1 - T )^{6} \) |
| good | 7 | \( 1 - 9 T^{2} + 59 T^{4} - 614 T^{6} + 59 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T - 2 T^{2} + 45 T^{3} - 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 16 T^{2} + 32 T^{4} + 1782 T^{6} + 32 p^{2} T^{8} - 16 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 60 T^{2} + 1760 T^{4} - 34586 T^{6} + 1760 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + T + 12 T^{2} - 47 T^{3} + 12 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 97 T^{2} + 4643 T^{4} - 133686 T^{6} + 4643 p^{2} T^{8} - 97 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 6 T + 71 T^{2} + 332 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - p T^{2} )^{6} \) |
| 41 | \( ( 1 - 6 T + 23 T^{2} - 20 T^{3} + 23 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 225 T^{2} + 22307 T^{4} - 1245350 T^{6} + 22307 p^{2} T^{8} - 225 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 176 T^{2} + 14112 T^{4} - 755006 T^{6} + 14112 p^{2} T^{8} - 176 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 157 T^{2} + 14507 T^{4} - 945006 T^{6} + 14507 p^{2} T^{8} - 157 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 2 T + 133 T^{2} + 276 T^{3} + 133 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 16 T + 248 T^{2} - 1962 T^{3} + 248 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 288 T^{2} + 40232 T^{4} - 3395714 T^{6} + 40232 p^{2} T^{8} - 288 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 9 T + 138 T^{2} - 1235 T^{3} + 138 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 205 T^{2} + 25427 T^{4} - 2278590 T^{6} + 25427 p^{2} T^{8} - 205 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 11 T + 96 T^{2} + 563 T^{3} + 96 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 4 p T^{2} + 55848 T^{4} - 5777546 T^{6} + 55848 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 29 T + 531 T^{2} + 5898 T^{3} + 531 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 444 T^{2} + 90312 T^{4} - 10990582 T^{6} + 90312 p^{2} T^{8} - 444 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.31858402971488320460418665568, −5.10907662013859404269781812493, −4.85289405987524021816122538441, −4.80462145462906018445902877862, −4.59957708624467858620712586220, −4.57518180656865574116441096227, −4.34704890597374620454112435075, −4.07560655381895995282978745950, −3.85536816313664361228109378132, −3.83717696812735753892781644990, −3.71518288851732628121666787720, −3.52017977232265757727018159209, −3.51210471927325030614819529792, −3.42449187422810904860881637858, −2.79782609462110289642465101269, −2.71205513838039692944580665886, −2.55562327694977164531846327964, −2.32532413187893770101005317513, −2.21988188673473671521923161394, −1.80768634455546148930607158416, −1.28640420678961987446190023845, −1.10873804122669826591092970611, −0.933250612832069360313266429736, −0.47980034374181565795987252548, −0.07154066185754190947194523075,
0.07154066185754190947194523075, 0.47980034374181565795987252548, 0.933250612832069360313266429736, 1.10873804122669826591092970611, 1.28640420678961987446190023845, 1.80768634455546148930607158416, 2.21988188673473671521923161394, 2.32532413187893770101005317513, 2.55562327694977164531846327964, 2.71205513838039692944580665886, 2.79782609462110289642465101269, 3.42449187422810904860881637858, 3.51210471927325030614819529792, 3.52017977232265757727018159209, 3.71518288851732628121666787720, 3.83717696812735753892781644990, 3.85536816313664361228109378132, 4.07560655381895995282978745950, 4.34704890597374620454112435075, 4.57518180656865574116441096227, 4.59957708624467858620712586220, 4.80462145462906018445902877862, 4.85289405987524021816122538441, 5.10907662013859404269781812493, 5.31858402971488320460418665568
Plot not available for L-functions of degree greater than 10.
