| L(s) = 1 | + 12·31-s + 8·37-s + 20·41-s + 8·43-s + 6·49-s − 12·53-s − 24·67-s − 8·71-s + 36·79-s + 32·83-s − 28·89-s + 40·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2.15·31-s + 1.31·37-s + 3.12·41-s + 1.21·43-s + 6/7·49-s − 1.64·53-s − 2.93·67-s − 0.949·71-s + 4.05·79-s + 3.51·83-s − 2.96·89-s + 3.86·107-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.030652260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.030652260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 47 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 11 T^{2} + 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 2 p T^{2} + 351 T^{4} - 1084 T^{6} + 351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 4527 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 4 T + 51 T^{2} - 40 T^{3} + 51 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 - 10 T + 87 T^{2} - 588 T^{3} + 87 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 4 T + 65 T^{2} - 216 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 7967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 59 | \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 4 T + p T^{2} )^{6} \) |
| 71 | \( ( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 16 T + 265 T^{2} - 2400 T^{3} + 265 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 14 T + 263 T^{2} + 2308 T^{3} + 263 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 24143 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.17271684919705743138056530015, −3.84516005590690091010073541561, −3.71334941716138969313287184318, −3.53133240976739450476386198701, −3.51735080117891841472162698646, −3.30980988675858050862024954871, −3.30434479186679446802700638066, −3.26271011873986072133740578230, −2.78477607201820678048999852324, −2.63552944886114835397068264812, −2.60498929409042344522251271358, −2.58930978898074878034386062334, −2.23317732001403656619691897218, −2.23232521015764167104602020700, −2.22543939261080394598991939710, −2.15795146686146262279957170182, −1.64409434111506852038969814190, −1.36345549334108916703780607496, −1.27848470901130704856010519012, −1.19001595821921610279136939980, −1.10242192830495995413111636945, −0.868366058513873293753944808825, −0.70353280188230069593065460679, −0.35354658926883641244876632085, −0.14945877585991323928488059560,
0.14945877585991323928488059560, 0.35354658926883641244876632085, 0.70353280188230069593065460679, 0.868366058513873293753944808825, 1.10242192830495995413111636945, 1.19001595821921610279136939980, 1.27848470901130704856010519012, 1.36345549334108916703780607496, 1.64409434111506852038969814190, 2.15795146686146262279957170182, 2.22543939261080394598991939710, 2.23232521015764167104602020700, 2.23317732001403656619691897218, 2.58930978898074878034386062334, 2.60498929409042344522251271358, 2.63552944886114835397068264812, 2.78477607201820678048999852324, 3.26271011873986072133740578230, 3.30434479186679446802700638066, 3.30980988675858050862024954871, 3.51735080117891841472162698646, 3.53133240976739450476386198701, 3.71334941716138969313287184318, 3.84516005590690091010073541561, 4.17271684919705743138056530015
Plot not available for L-functions of degree greater than 10.
