L(s) = 1 | + 4·3-s − 4·5-s + 4·7-s − 6·11-s + 2·13-s − 16·15-s − 10·17-s + 6·19-s + 16·21-s + 14·23-s − 6·25-s − 20·27-s + 6·29-s − 4·31-s − 24·33-s − 16·35-s − 4·37-s + 8·39-s − 8·41-s + 26·43-s + 24·47-s − 3·49-s − 40·51-s − 8·53-s + 24·55-s + 24·57-s + 4·59-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 1.78·5-s + 1.51·7-s − 1.80·11-s + 0.554·13-s − 4.13·15-s − 2.42·17-s + 1.37·19-s + 3.49·21-s + 2.91·23-s − 6/5·25-s − 3.84·27-s + 1.11·29-s − 0.718·31-s − 4.17·33-s − 2.70·35-s − 0.657·37-s + 1.28·39-s − 1.24·41-s + 3.96·43-s + 3.50·47-s − 3/7·49-s − 5.60·51-s − 1.09·53-s + 3.23·55-s + 3.17·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.521290438\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.521290438\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( ( 1 + T )^{6} \) |
| 19 | \( ( 1 - T )^{6} \) |
good | 3 | \( 1 - 4 T + 16 T^{2} - 44 T^{3} + 37 p T^{4} - 224 T^{5} + 431 T^{6} - 224 p T^{7} + 37 p^{3} T^{8} - 44 p^{3} T^{9} + 16 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 + 4 T + 22 T^{2} + 54 T^{3} + 39 p T^{4} + 398 T^{5} + 1181 T^{6} + 398 p T^{7} + 39 p^{3} T^{8} + 54 p^{3} T^{9} + 22 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 4 T + 19 T^{2} - 24 T^{3} + 26 T^{4} + 270 T^{5} - 634 T^{6} + 270 p T^{7} + 26 p^{2} T^{8} - 24 p^{3} T^{9} + 19 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 2 T + 11 T^{2} - 40 T^{3} - 64 T^{4} + 560 T^{5} - 150 p T^{6} + 560 p T^{7} - 64 p^{2} T^{8} - 40 p^{3} T^{9} + 11 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 10 T + 96 T^{2} + 626 T^{3} + 3767 T^{4} + 18724 T^{5} + 83600 T^{6} + 18724 p T^{7} + 3767 p^{2} T^{8} + 626 p^{3} T^{9} + 96 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 14 T + 143 T^{2} - 50 p T^{3} + 353 p T^{4} - 47280 T^{5} + 242210 T^{6} - 47280 p T^{7} + 353 p^{3} T^{8} - 50 p^{4} T^{9} + 143 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 6 T + 109 T^{2} - 444 T^{3} + 5772 T^{4} - 21092 T^{5} + 211254 T^{6} - 21092 p T^{7} + 5772 p^{2} T^{8} - 444 p^{3} T^{9} + 109 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 4 T + 30 T^{2} - 280 T^{3} + 145 T^{4} + 360 T^{5} + 93811 T^{6} + 360 p T^{7} + 145 p^{2} T^{8} - 280 p^{3} T^{9} + 30 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 4 T + 27 T^{2} - 172 T^{3} + 539 T^{4} - 7624 T^{5} + 18418 T^{6} - 7624 p T^{7} + 539 p^{2} T^{8} - 172 p^{3} T^{9} + 27 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 + 8 T + 149 T^{2} + 1084 T^{3} + 11878 T^{4} + 69308 T^{5} + 606740 T^{6} + 69308 p T^{7} + 11878 p^{2} T^{8} + 1084 p^{3} T^{9} + 149 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 26 T + 421 T^{2} - 4824 T^{3} + 46208 T^{4} - 372090 T^{5} + 2638016 T^{6} - 372090 p T^{7} + 46208 p^{2} T^{8} - 4824 p^{3} T^{9} + 421 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 24 T + 346 T^{2} - 3512 T^{3} + 27311 T^{4} - 184288 T^{5} + 1213804 T^{6} - 184288 p T^{7} + 27311 p^{2} T^{8} - 3512 p^{3} T^{9} + 346 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 8 T + 246 T^{2} + 1672 T^{3} + 28743 T^{4} + 159504 T^{5} + 1944308 T^{6} + 159504 p T^{7} + 28743 p^{2} T^{8} + 1672 p^{3} T^{9} + 246 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 4 T + 243 T^{2} - 772 T^{3} + 27863 T^{4} - 68336 T^{5} + 2001626 T^{6} - 68336 p T^{7} + 27863 p^{2} T^{8} - 772 p^{3} T^{9} + 243 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 6 T + 104 T^{2} + 510 T^{3} + 9759 T^{4} + 33028 T^{5} + 461984 T^{6} + 33028 p T^{7} + 9759 p^{2} T^{8} + 510 p^{3} T^{9} + 104 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 44 T + 1062 T^{2} - 18104 T^{3} + 239877 T^{4} - 2583108 T^{5} + 23113947 T^{6} - 2583108 p T^{7} + 239877 p^{2} T^{8} - 18104 p^{3} T^{9} + 1062 p^{4} T^{10} - 44 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 16 T + 468 T^{2} - 5336 T^{3} + 86087 T^{4} - 733296 T^{5} + 8224119 T^{6} - 733296 p T^{7} + 86087 p^{2} T^{8} - 5336 p^{3} T^{9} + 468 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 12 T + 162 T^{2} - 1468 T^{3} + 17471 T^{4} - 113304 T^{5} + 1131548 T^{6} - 113304 p T^{7} + 17471 p^{2} T^{8} - 1468 p^{3} T^{9} + 162 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 6 T + 288 T^{2} + 2546 T^{3} + 40159 T^{4} + 404524 T^{5} + 3723232 T^{6} + 404524 p T^{7} + 40159 p^{2} T^{8} + 2546 p^{3} T^{9} + 288 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 28 T + 615 T^{2} - 8888 T^{3} + 119678 T^{4} - 1285446 T^{5} + 13052918 T^{6} - 1285446 p T^{7} + 119678 p^{2} T^{8} - 8888 p^{3} T^{9} + 615 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 295 T^{2} - 704 T^{3} + 42883 T^{4} - 137280 T^{5} + 4439098 T^{6} - 137280 p T^{7} + 42883 p^{2} T^{8} - 704 p^{3} T^{9} + 295 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 4 T + 323 T^{2} - 2240 T^{3} + 58575 T^{4} - 364268 T^{5} + 7255562 T^{6} - 364268 p T^{7} + 58575 p^{2} T^{8} - 2240 p^{3} T^{9} + 323 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.52533193933686915062743443921, −4.16549966795268995290795807357, −4.07816011197735133719052690780, −3.79161947828347454648720121541, −3.77341782049303507541839090357, −3.73342393259584133202822137080, −3.66552387152147528298656392047, −3.35055381531440227400464055257, −3.24064191224085637943820577349, −3.11490106310496493211483630729, −3.04378902255460495344846114525, −2.73844885377036331350869085990, −2.58031720376365936504828025293, −2.42572335263832001436058762522, −2.37210925642116337463351732076, −2.26268208172427283803297899565, −2.14158926645011181949343786268, −2.04215450532832161273446232016, −1.81247507996602143011564296650, −1.29767337453609823362529986453, −1.17985019410955671794328576845, −0.830880227302575651546588562832, −0.67372576599023211874938158282, −0.59341200883244010662176331745, −0.20601475513121320509638761630,
0.20601475513121320509638761630, 0.59341200883244010662176331745, 0.67372576599023211874938158282, 0.830880227302575651546588562832, 1.17985019410955671794328576845, 1.29767337453609823362529986453, 1.81247507996602143011564296650, 2.04215450532832161273446232016, 2.14158926645011181949343786268, 2.26268208172427283803297899565, 2.37210925642116337463351732076, 2.42572335263832001436058762522, 2.58031720376365936504828025293, 2.73844885377036331350869085990, 3.04378902255460495344846114525, 3.11490106310496493211483630729, 3.24064191224085637943820577349, 3.35055381531440227400464055257, 3.66552387152147528298656392047, 3.73342393259584133202822137080, 3.77341782049303507541839090357, 3.79161947828347454648720121541, 4.07816011197735133719052690780, 4.16549966795268995290795807357, 4.52533193933686915062743443921
Plot not available for L-functions of degree greater than 10.