| L(s) = 1 | − 3·3-s + 3·5-s + 3·9-s + 3·11-s − 6·13-s − 9·15-s − 6·17-s − 3·19-s + 3·23-s + 3·25-s + 2·27-s + 12·29-s − 12·31-s − 9·33-s − 3·37-s + 18·39-s − 18·41-s + 9·45-s − 3·47-s + 6·49-s + 18·51-s + 15·53-s + 9·55-s + 9·57-s − 30·59-s − 18·65-s − 9·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 9-s + 0.904·11-s − 1.66·13-s − 2.32·15-s − 1.45·17-s − 0.688·19-s + 0.625·23-s + 3/5·25-s + 0.384·27-s + 2.22·29-s − 2.15·31-s − 1.56·33-s − 0.493·37-s + 2.88·39-s − 2.81·41-s + 1.34·45-s − 0.437·47-s + 6/7·49-s + 2.52·51-s + 2.06·53-s + 1.21·55-s + 1.19·57-s − 3.90·59-s − 2.23·65-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03813358587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03813358587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T + T^{2} )^{3} \) |
| 5 | \( ( 1 - T + T^{2} )^{3} \) |
| 7 | \( 1 - 6 T^{2} - 4 T^{3} - 6 p T^{4} + p^{3} T^{6} \) |
| good | 11 | \( 1 - 3 T + 12 T^{2} - 111 T^{3} + 222 T^{4} - 741 T^{5} + 5074 T^{6} - 741 p T^{7} + 222 p^{2} T^{8} - 111 p^{3} T^{9} + 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 3 T + 24 T^{2} + 81 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 + 6 T + 39 T^{2} + 214 T^{3} + 756 T^{4} + 2694 T^{5} + 12725 T^{6} + 2694 p T^{7} + 756 p^{2} T^{8} + 214 p^{3} T^{9} + 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 7 T + p T^{2} )^{3}( 1 + 8 T + p T^{2} )^{3} \) |
| 23 | \( 1 - 3 T - 36 T^{2} + 97 T^{3} + 642 T^{4} - 597 T^{5} - 13546 T^{6} - 597 p T^{7} + 642 p^{2} T^{8} + 97 p^{3} T^{9} - 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 6 T + 81 T^{2} - 300 T^{3} + 81 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 12 T + 78 T^{2} + 204 T^{3} - 1218 T^{4} - 18168 T^{5} - 127402 T^{6} - 18168 p T^{7} - 1218 p^{2} T^{8} + 204 p^{3} T^{9} + 78 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 3 T - 15 T^{2} + 390 T^{3} + 165 T^{4} - 5037 T^{5} + 92954 T^{6} - 5037 p T^{7} + 165 p^{2} T^{8} + 390 p^{3} T^{9} - 15 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 9 T + 3 p T^{2} + 680 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 90 T^{2} + 88 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 3 T - 60 T^{2} - 637 T^{3} + 252 T^{4} + 16287 T^{5} + 142082 T^{6} + 16287 p T^{7} + 252 p^{2} T^{8} - 637 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 15 T + 6 T^{2} + 7 T^{3} + 11664 T^{4} - 54399 T^{5} - 166888 T^{6} - 54399 p T^{7} + 11664 p^{2} T^{8} + 7 p^{3} T^{9} + 6 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 30 T + 441 T^{2} + 5010 T^{3} + 52200 T^{4} + 489990 T^{5} + 4032079 T^{6} + 489990 p T^{7} + 52200 p^{2} T^{8} + 5010 p^{3} T^{9} + 441 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 67 | \( 1 - 162 T^{2} + 176 T^{3} + 15390 T^{4} - 14256 T^{5} - 1149078 T^{6} - 14256 p T^{7} + 15390 p^{2} T^{8} + 176 p^{3} T^{9} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 12 T + 27 T^{2} - 396 T^{3} + 27 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 3 T - 36 T^{2} + 887 T^{3} - 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )( 1 + 21 T + 123 T^{2} + 266 T^{3} + 123 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 79 | \( 1 + 6 T - 114 T^{2} - 932 T^{3} + 5154 T^{4} + 32646 T^{5} - 177042 T^{6} + 32646 p T^{7} + 5154 p^{2} T^{8} - 932 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 6 T + 243 T^{2} - 948 T^{3} + 243 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 177 T^{2} + 584 T^{3} + 15576 T^{4} - 51684 T^{5} - 1293079 T^{6} - 51684 p T^{7} + 15576 p^{2} T^{8} + 584 p^{3} T^{9} - 177 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 8 T + p T^{2} )^{6} \) |
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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
−5.57587757267864693719473669944, −5.14703525009035619164560642629, −5.10742875381043783037825335929, −4.94326358015456557952512389600, −4.81704341454822452068348368224, −4.77739823862750259314047641792, −4.45876845610637734333350449876, −4.32045508774351851217017731541, −4.30415156761521899372693141184, −3.95220296303094546051876300136, −3.83240029082583035434598238795, −3.43860768930661203081683556792, −3.19004792999902980564227393907, −3.15468598154034647288242839700, −2.92310045113326080228739676301, −2.84837170066588427473842552769, −2.29689182130536587221841021023, −2.20754931358730762982182936348, −2.16180235773029925661959994095, −1.75022989331057447238414283647, −1.67804961283783817152710506802, −1.24280034608058353820256524173, −1.17372716660844993512193863955, −0.51196472929046789237511805617, −0.05182487095869677287064870680,
0.05182487095869677287064870680, 0.51196472929046789237511805617, 1.17372716660844993512193863955, 1.24280034608058353820256524173, 1.67804961283783817152710506802, 1.75022989331057447238414283647, 2.16180235773029925661959994095, 2.20754931358730762982182936348, 2.29689182130536587221841021023, 2.84837170066588427473842552769, 2.92310045113326080228739676301, 3.15468598154034647288242839700, 3.19004792999902980564227393907, 3.43860768930661203081683556792, 3.83240029082583035434598238795, 3.95220296303094546051876300136, 4.30415156761521899372693141184, 4.32045508774351851217017731541, 4.45876845610637734333350449876, 4.77739823862750259314047641792, 4.81704341454822452068348368224, 4.94326358015456557952512389600, 5.10742875381043783037825335929, 5.14703525009035619164560642629, 5.57587757267864693719473669944
Plot not available for L-functions of degree greater than 10.
