| L(s) = 1 | + 7·5-s − 6·7-s − 2·11-s − 6·13-s − 4·17-s − 8·19-s + 7·23-s + 28·25-s + 2·29-s − 8·31-s − 42·35-s − 16·37-s + 2·41-s − 10·43-s − 8·47-s + 3·49-s − 10·53-s − 14·55-s − 24·59-s + 8·61-s − 42·65-s − 20·67-s − 8·71-s + 2·73-s + 12·77-s − 2·79-s − 22·83-s + ⋯ |
| L(s) = 1 | + 3.13·5-s − 2.26·7-s − 0.603·11-s − 1.66·13-s − 0.970·17-s − 1.83·19-s + 1.45·23-s + 28/5·25-s + 0.371·29-s − 1.43·31-s − 7.09·35-s − 2.63·37-s + 0.312·41-s − 1.52·43-s − 1.16·47-s + 3/7·49-s − 1.37·53-s − 1.88·55-s − 3.12·59-s + 1.02·61-s − 5.20·65-s − 2.44·67-s − 0.949·71-s + 0.234·73-s + 1.36·77-s − 0.225·79-s − 2.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{14} \cdot 5^{7} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{14} \cdot 5^{7} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 - T )^{7} \) |
| 23 | \( ( 1 - T )^{7} \) |
| good | 7 | \( 1 + 6 T + 33 T^{2} + 118 T^{3} + 8 p^{2} T^{4} + 1174 T^{5} + 3280 T^{6} + 9244 T^{7} + 3280 p T^{8} + 1174 p^{2} T^{9} + 8 p^{5} T^{10} + 118 p^{4} T^{11} + 33 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) |
| 11 | \( 1 + 2 T + 27 T^{2} + 4 T^{3} + 347 T^{4} - 74 T^{5} + 6161 T^{6} + 3112 T^{7} + 6161 p T^{8} - 74 p^{2} T^{9} + 347 p^{3} T^{10} + 4 p^{4} T^{11} + 27 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 13 | \( 1 + 6 T + 33 T^{2} + 220 T^{3} + 1191 T^{4} + 4386 T^{5} + 19575 T^{6} + 80376 T^{7} + 19575 p T^{8} + 4386 p^{2} T^{9} + 1191 p^{3} T^{10} + 220 p^{4} T^{11} + 33 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) |
| 17 | \( 1 + 4 T + 31 T^{2} + 44 T^{3} + 188 T^{4} + 428 T^{5} + 5748 T^{6} + 25736 T^{7} + 5748 p T^{8} + 428 p^{2} T^{9} + 188 p^{3} T^{10} + 44 p^{4} T^{11} + 31 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) |
| 19 | \( 1 + 8 T + 59 T^{2} + 396 T^{3} + 2091 T^{4} + 10760 T^{5} + 53985 T^{6} + 229640 T^{7} + 53985 p T^{8} + 10760 p^{2} T^{9} + 2091 p^{3} T^{10} + 396 p^{4} T^{11} + 59 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) |
| 29 | \( 1 - 2 T + 105 T^{2} - 246 T^{3} + 5960 T^{4} - 13694 T^{5} + 243530 T^{6} - 477700 T^{7} + 243530 p T^{8} - 13694 p^{2} T^{9} + 5960 p^{3} T^{10} - 246 p^{4} T^{11} + 105 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 31 | \( 1 + 8 T + 55 T^{2} + 264 T^{3} + 2080 T^{4} + 8824 T^{5} + 51350 T^{6} + 222192 T^{7} + 51350 p T^{8} + 8824 p^{2} T^{9} + 2080 p^{3} T^{10} + 264 p^{4} T^{11} + 55 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) |
| 37 | \( 1 + 16 T + 219 T^{2} + 1918 T^{3} + 14316 T^{4} + 90054 T^{5} + 13656 p T^{6} + 3098944 T^{7} + 13656 p^{2} T^{8} + 90054 p^{2} T^{9} + 14316 p^{3} T^{10} + 1918 p^{4} T^{11} + 219 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) |
| 41 | \( 1 - 2 T + 97 T^{2} - 490 T^{3} + 7340 T^{4} - 35154 T^{5} + 389214 T^{6} - 1901076 T^{7} + 389214 p T^{8} - 35154 p^{2} T^{9} + 7340 p^{3} T^{10} - 490 p^{4} T^{11} + 97 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 43 | \( 1 + 10 T + 137 T^{2} + 1404 T^{3} + 12929 T^{4} + 101862 T^{5} + 740097 T^{6} + 5407112 T^{7} + 740097 p T^{8} + 101862 p^{2} T^{9} + 12929 p^{3} T^{10} + 1404 p^{4} T^{11} + 137 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) |
| 47 | \( 1 + 8 T + 167 T^{2} + 764 T^{3} + 8075 T^{4} - 4920 T^{5} + 8277 T^{6} - 2172568 T^{7} + 8277 p T^{8} - 4920 p^{2} T^{9} + 8075 p^{3} T^{10} + 764 p^{4} T^{11} + 167 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) |
| 53 | \( 1 + 10 T + 227 T^{2} + 1440 T^{3} + 420 p T^{4} + 103448 T^{5} + 27076 p T^{6} + 5542636 T^{7} + 27076 p^{2} T^{8} + 103448 p^{2} T^{9} + 420 p^{4} T^{10} + 1440 p^{4} T^{11} + 227 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) |
| 59 | \( 1 + 24 T + 439 T^{2} + 5646 T^{3} + 62028 T^{4} + 562974 T^{5} + 4793650 T^{6} + 37001200 T^{7} + 4793650 p T^{8} + 562974 p^{2} T^{9} + 62028 p^{3} T^{10} + 5646 p^{4} T^{11} + 439 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) |
| 61 | \( 1 - 8 T + 149 T^{2} - 940 T^{3} + 15171 T^{4} - 92440 T^{5} + 1152647 T^{6} - 6156392 T^{7} + 1152647 p T^{8} - 92440 p^{2} T^{9} + 15171 p^{3} T^{10} - 940 p^{4} T^{11} + 149 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) |
| 67 | \( 1 + 20 T + 293 T^{2} + 3462 T^{3} + 36580 T^{4} + 343950 T^{5} + 3091200 T^{6} + 26654016 T^{7} + 3091200 p T^{8} + 343950 p^{2} T^{9} + 36580 p^{3} T^{10} + 3462 p^{4} T^{11} + 293 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) |
| 71 | \( 1 + 8 T + 207 T^{2} + 982 T^{3} + 272 p T^{4} + 33686 T^{5} + 15266 p T^{6} - 537032 T^{7} + 15266 p^{2} T^{8} + 33686 p^{2} T^{9} + 272 p^{4} T^{10} + 982 p^{4} T^{11} + 207 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) |
| 73 | \( 1 - 2 T + 185 T^{2} - 68 T^{3} + 25315 T^{4} - 14086 T^{5} + 2389155 T^{6} - 335144 T^{7} + 2389155 p T^{8} - 14086 p^{2} T^{9} + 25315 p^{3} T^{10} - 68 p^{4} T^{11} + 185 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 79 | \( 1 + 2 T + 281 T^{2} + 692 T^{3} + 42565 T^{4} + 120670 T^{5} + 4537101 T^{6} + 12474200 T^{7} + 4537101 p T^{8} + 120670 p^{2} T^{9} + 42565 p^{3} T^{10} + 692 p^{4} T^{11} + 281 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 83 | \( 1 + 22 T + 553 T^{2} + 8176 T^{3} + 127120 T^{4} + 1444144 T^{5} + 16806392 T^{6} + 151937092 T^{7} + 16806392 p T^{8} + 1444144 p^{2} T^{9} + 127120 p^{3} T^{10} + 8176 p^{4} T^{11} + 553 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) |
| 89 | \( 1 + 16 T + 347 T^{2} + 3840 T^{3} + 58753 T^{4} + 571120 T^{5} + 7185331 T^{6} + 59722240 T^{7} + 7185331 p T^{8} + 571120 p^{2} T^{9} + 58753 p^{3} T^{10} + 3840 p^{4} T^{11} + 347 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) |
| 97 | \( 1 - 4 T + 439 T^{2} - 1848 T^{3} + 97605 T^{4} - 391100 T^{5} + 13827779 T^{6} - 48684048 T^{7} + 13827779 p T^{8} - 391100 p^{2} T^{9} + 97605 p^{3} T^{10} - 1848 p^{4} T^{11} + 439 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.83535495258087048162430432555, −3.81193226041421672985285179742, −3.68839875273363422511376726731, −3.67645651818743697749296613655, −3.25047270140337722833042008825, −3.11444443979979573451543209715, −3.11142428144872488647311875728, −3.11006915028375604249485729118, −2.95423908732794125771663264266, −2.88134595511014844243387105324, −2.67986362879437695633631776614, −2.57165148170900513437821832899, −2.44027242961891230468428687423, −2.32216473244915260411562862759, −2.31085032481937241033234797353, −2.17618457851353492721561722587, −2.10172298543623814778948795229, −1.95025456946046942060788146022, −1.55090063688615433536504416990, −1.42093274705204756544566029601, −1.38221587786334131538601221574, −1.34570275285941873900661541089, −1.21152083180771539822521474062, −1.19406777504623485965843193053, −1.15086635764817324235856450949, 0, 0, 0, 0, 0, 0, 0,
1.15086635764817324235856450949, 1.19406777504623485965843193053, 1.21152083180771539822521474062, 1.34570275285941873900661541089, 1.38221587786334131538601221574, 1.42093274705204756544566029601, 1.55090063688615433536504416990, 1.95025456946046942060788146022, 2.10172298543623814778948795229, 2.17618457851353492721561722587, 2.31085032481937241033234797353, 2.32216473244915260411562862759, 2.44027242961891230468428687423, 2.57165148170900513437821832899, 2.67986362879437695633631776614, 2.88134595511014844243387105324, 2.95423908732794125771663264266, 3.11006915028375604249485729118, 3.11142428144872488647311875728, 3.11444443979979573451543209715, 3.25047270140337722833042008825, 3.67645651818743697749296613655, 3.68839875273363422511376726731, 3.81193226041421672985285179742, 3.83535495258087048162430432555
Plot not available for L-functions of degree greater than 10.
