L(s) = 1 | − 5-s + 2·7-s − 5·11-s − 4·13-s + 4·17-s − 3·19-s − 14·23-s + 3·25-s − 10·29-s + 2·31-s − 2·35-s + 24·41-s − 18·43-s + 9·47-s + 2·49-s − 6·53-s + 5·55-s + 5·59-s − 7·61-s + 4·65-s + 16·67-s + 22·71-s + 73-s − 10·77-s + 8·79-s + 34·83-s − 4·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.755·7-s − 1.50·11-s − 1.10·13-s + 0.970·17-s − 0.688·19-s − 2.91·23-s + 3/5·25-s − 1.85·29-s + 0.359·31-s − 0.338·35-s + 3.74·41-s − 2.74·43-s + 1.31·47-s + 2/7·49-s − 0.824·53-s + 0.674·55-s + 0.650·59-s − 0.896·61-s + 0.496·65-s + 1.95·67-s + 2.61·71-s + 0.117·73-s − 1.13·77-s + 0.900·79-s + 3.73·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.678758949\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.678758949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2 T + 2 T^{2} + 19 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
good | 5 | \( 1 + T - 2 T^{2} - 7 p T^{3} - p^{2} T^{4} + 52 T^{5} + 541 T^{6} + 52 p T^{7} - p^{4} T^{8} - 7 p^{4} T^{9} - 2 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 5 T + 4 T^{2} + p T^{3} - 25 T^{4} - 718 T^{5} - 3005 T^{6} - 718 p T^{7} - 25 p^{2} T^{8} + p^{4} T^{9} + 4 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 2 T + 28 T^{2} + 31 T^{3} + 28 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 4 T - 20 T^{2} + 146 T^{3} + 104 T^{4} - 1480 T^{5} + 4195 T^{6} - 1480 p T^{7} + 104 p^{2} T^{8} + 146 p^{3} T^{9} - 20 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 54 p T^{7} - 153 p^{2} T^{8} - 67 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 14 T + 88 T^{2} + 350 T^{3} + 1046 T^{4} + 602 T^{5} - 10061 T^{6} + 602 p T^{7} + 1046 p^{2} T^{8} + 350 p^{3} T^{9} + 88 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 5 T + 75 T^{2} + 227 T^{3} + 75 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 2 T - 78 T^{2} + 42 T^{3} + 3976 T^{4} - 200 T^{5} - 142097 T^{6} - 200 p T^{7} + 3976 p^{2} T^{8} + 42 p^{3} T^{9} - 78 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 - 54 T^{2} - 274 T^{3} + 918 T^{4} + 7398 T^{5} + 12183 T^{6} + 7398 p T^{7} + 918 p^{2} T^{8} - 274 p^{3} T^{9} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 12 T + 132 T^{2} - 903 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 9 T + 99 T^{2} + 493 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 9 T - 6 T^{2} + 9 p T^{3} - 1947 T^{4} + 1260 T^{5} + 36583 T^{6} + 1260 p T^{7} - 1947 p^{2} T^{8} + 9 p^{4} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 6 T - 24 T^{2} + 222 T^{3} + 6 T^{4} - 17166 T^{5} + 10591 T^{6} - 17166 p T^{7} + 6 p^{2} T^{8} + 222 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 5 T - 134 T^{2} + 223 T^{3} + 13355 T^{4} - 4526 T^{5} - 908765 T^{6} - 4526 p T^{7} + 13355 p^{2} T^{8} + 223 p^{3} T^{9} - 134 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 7 T - 45 T^{2} + 84 T^{3} + 1093 T^{4} - 27251 T^{5} - 184994 T^{6} - 27251 p T^{7} + 1093 p^{2} T^{8} + 84 p^{3} T^{9} - 45 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 16 T - 4 T^{2} + 234 T^{3} + 15028 T^{4} - 85604 T^{5} - 250049 T^{6} - 85604 p T^{7} + 15028 p^{2} T^{8} + 234 p^{3} T^{9} - 4 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 11 T + 141 T^{2} - 1373 T^{3} + 141 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - T - 136 T^{2} + 477 T^{3} + 8461 T^{4} - 27386 T^{5} - 503183 T^{6} - 27386 p T^{7} + 8461 p^{2} T^{8} + 477 p^{3} T^{9} - 136 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 8 T - 18 T^{2} + 126 T^{3} - 2882 T^{4} + 21658 T^{5} + 208339 T^{6} + 21658 p T^{7} - 2882 p^{2} T^{8} + 126 p^{3} T^{9} - 18 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 17 T + 333 T^{2} - 2921 T^{3} + 333 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 3 T - 33 T^{2} - 1704 T^{3} + 393 T^{4} + 31803 T^{5} + 1804174 T^{6} + 31803 p T^{7} + 393 p^{2} T^{8} - 1704 p^{3} T^{9} - 33 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 14 T + 251 T^{2} - 2660 T^{3} + 251 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−5.29506620147120892830621872982, −5.25984115032971796425899837813, −5.21656546488957749465779683935, −5.01862145242924808445899959582, −4.96500989263169547588263187002, −4.69190709367249105029218154987, −4.60743350287344586503040100912, −4.31021009595105012611833322482, −4.03751199991665491548538279779, −3.97622422468823925535762190792, −3.75535235404111042530251109991, −3.61613245064773058389259214778, −3.52263208261494885489982937587, −3.31180128413584180940656848660, −2.91635846889312121872544117964, −2.65599536862410957808295654032, −2.62351203008984377906415211382, −2.15949552201239371651325248909, −2.14043023330103465195535916081, −2.09160014091316853000284898765, −1.79236073353234755384242433461, −1.52818415336946036701435313843, −0.73640916115462606665485275938, −0.71169917092970086290171306817, −0.42184164588472040191055514610,
0.42184164588472040191055514610, 0.71169917092970086290171306817, 0.73640916115462606665485275938, 1.52818415336946036701435313843, 1.79236073353234755384242433461, 2.09160014091316853000284898765, 2.14043023330103465195535916081, 2.15949552201239371651325248909, 2.62351203008984377906415211382, 2.65599536862410957808295654032, 2.91635846889312121872544117964, 3.31180128413584180940656848660, 3.52263208261494885489982937587, 3.61613245064773058389259214778, 3.75535235404111042530251109991, 3.97622422468823925535762190792, 4.03751199991665491548538279779, 4.31021009595105012611833322482, 4.60743350287344586503040100912, 4.69190709367249105029218154987, 4.96500989263169547588263187002, 5.01862145242924808445899959582, 5.21656546488957749465779683935, 5.25984115032971796425899837813, 5.29506620147120892830621872982
Plot not available for L-functions of degree greater than 10.