| L(s) = 1 | + 3·3-s + 6·5-s − 3·7-s + 4·8-s + 3·9-s + 3·13-s + 18·15-s − 12·19-s − 9·21-s + 12·24-s + 21·25-s − 2·27-s − 6·29-s + 6·31-s − 18·35-s − 6·37-s + 9·39-s + 24·40-s − 9·43-s + 18·45-s − 24·47-s + 18·49-s − 12·56-s − 36·57-s + 6·59-s + 3·61-s − 9·63-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2.68·5-s − 1.13·7-s + 1.41·8-s + 9-s + 0.832·13-s + 4.64·15-s − 2.75·19-s − 1.96·21-s + 2.44·24-s + 21/5·25-s − 0.384·27-s − 1.11·29-s + 1.07·31-s − 3.04·35-s − 0.986·37-s + 1.44·39-s + 3.79·40-s − 1.37·43-s + 2.68·45-s − 3.50·47-s + 18/7·49-s − 1.60·56-s − 4.76·57-s + 0.781·59-s + 0.384·61-s − 1.13·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.231866160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.231866160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - T + T^{2} )^{3} \) |
| 5 | \( ( 1 - T )^{6} \) |
| 13 | \( 1 - 3 T - 6 T^{2} + 83 T^{3} - 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( ( 1 - p T^{3} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 + 3 T - 9 T^{2} - 24 T^{3} + 81 T^{4} + 93 T^{5} - 502 T^{6} + 93 p T^{7} + 81 p^{2} T^{8} - 24 p^{3} T^{9} - 9 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 9 T^{2} + 32 T^{3} - 18 T^{4} - 144 T^{5} + 2027 T^{6} - 144 p T^{7} - 18 p^{2} T^{8} + 32 p^{3} T^{9} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 9 T^{2} - 196 T^{3} - 72 T^{4} + 882 T^{5} + 18053 T^{6} + 882 p T^{7} - 72 p^{2} T^{8} - 196 p^{3} T^{9} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 12 T + 51 T^{2} + 196 T^{3} + 1362 T^{4} + 4704 T^{5} + 9003 T^{6} + 4704 p T^{7} + 1362 p^{2} T^{8} + 196 p^{3} T^{9} + 51 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 21 T^{2} + 192 T^{3} - 42 T^{4} - 2016 T^{5} + 23407 T^{6} - 2016 p T^{7} - 42 p^{2} T^{8} + 192 p^{3} T^{9} - 21 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 + 6 T - 51 T^{2} - 146 T^{3} + 3042 T^{4} + 132 p T^{5} - 89443 T^{6} + 132 p^{2} T^{7} + 3042 p^{2} T^{8} - 146 p^{3} T^{9} - 51 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 3 T + 177 T^{3} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 6 T - 39 T^{2} - 454 T^{3} + 138 T^{4} + 8814 T^{5} + 48201 T^{6} + 8814 p T^{7} + 138 p^{2} T^{8} - 454 p^{3} T^{9} - 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 99 T^{2} + 52 T^{3} + 5742 T^{4} - 2574 T^{5} - 263323 T^{6} - 2574 p T^{7} + 5742 p^{2} T^{8} + 52 p^{3} T^{9} - 99 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 + 9 T - 63 T^{2} - 230 T^{3} + 7677 T^{4} + 15057 T^{5} - 300234 T^{6} + 15057 p T^{7} + 7677 p^{2} T^{8} - 230 p^{3} T^{9} - 63 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 12 T + 135 T^{2} + 922 T^{3} + 135 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 135 T^{2} + 16 T^{3} + 135 p T^{4} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 6 T - 129 T^{2} + 398 T^{3} + 13326 T^{4} - 18576 T^{5} - 839197 T^{6} - 18576 p T^{7} + 13326 p^{2} T^{8} + 398 p^{3} T^{9} - 129 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 3 T - 153 T^{2} + 112 T^{3} + 15465 T^{4} - 309 T^{5} - 1091010 T^{6} - 309 p T^{7} + 15465 p^{2} T^{8} + 112 p^{3} T^{9} - 153 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 9 T - 39 T^{2} - 630 T^{3} - 1335 T^{4} - 1719 T^{5} + 4142 T^{6} - 1719 p T^{7} - 1335 p^{2} T^{8} - 630 p^{3} T^{9} - 39 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 12 T - 21 T^{2} + 64 T^{3} + 3246 T^{4} + 52038 T^{5} - 898573 T^{6} + 52038 p T^{7} + 3246 p^{2} T^{8} + 64 p^{3} T^{9} - 21 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 21 T + 312 T^{2} - 2977 T^{3} + 312 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 3 T + 144 T^{2} + 111 T^{3} + 144 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 18 T + 309 T^{2} + 2820 T^{3} + 309 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 30 T + 345 T^{2} + 4206 T^{3} + 71970 T^{4} + 727140 T^{5} + 5580097 T^{6} + 727140 p T^{7} + 71970 p^{2} T^{8} + 4206 p^{3} T^{9} + 345 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 15 T - 57 T^{2} - 412 T^{3} + 17355 T^{4} - 24963 T^{5} - 2618466 T^{6} - 24963 p T^{7} + 17355 p^{2} T^{8} - 412 p^{3} T^{9} - 57 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.93645778300608959580290948255, −6.67094993085870160368038606220, −6.51861711369180711654084773718, −6.43479523708033768279597654776, −6.17822657232342866107092436665, −6.00063344088715188927538875985, −5.85459900438395986644154429864, −5.41801729938012026832166185171, −5.38970456860258875392726282673, −5.36961521232894928434742895297, −4.80640621050534447526874289051, −4.56088338955917465878227500055, −4.40593898816852424758953562828, −4.36377907484198883884988239255, −3.90648627320598093745059558695, −3.69101048014574956963407307250, −3.29316567301027369179350186713, −3.07908322064280282963263617828, −3.07680651255132390136709340196, −2.72041017923781110130181404337, −2.02846324427282973056080871056, −1.96018599622287071952816927742, −1.93449852983941611049713545056, −1.87971822295181699323293477405, −0.983423377660796407212717403871,
0.983423377660796407212717403871, 1.87971822295181699323293477405, 1.93449852983941611049713545056, 1.96018599622287071952816927742, 2.02846324427282973056080871056, 2.72041017923781110130181404337, 3.07680651255132390136709340196, 3.07908322064280282963263617828, 3.29316567301027369179350186713, 3.69101048014574956963407307250, 3.90648627320598093745059558695, 4.36377907484198883884988239255, 4.40593898816852424758953562828, 4.56088338955917465878227500055, 4.80640621050534447526874289051, 5.36961521232894928434742895297, 5.38970456860258875392726282673, 5.41801729938012026832166185171, 5.85459900438395986644154429864, 6.00063344088715188927538875985, 6.17822657232342866107092436665, 6.43479523708033768279597654776, 6.51861711369180711654084773718, 6.67094993085870160368038606220, 6.93645778300608959580290948255
Plot not available for L-functions of degree greater than 10.
