| L(s) = 1 | − 5·5-s − 3·7-s + 2·11-s − 3·13-s + 24·17-s + 6·19-s + 17·25-s + 29-s − 3·31-s + 15·35-s − 6·37-s − 22·41-s − 3·43-s + 9·47-s + 3·49-s + 36·53-s − 10·55-s + 9·59-s + 6·61-s + 15·65-s + 18·71-s + 6·73-s − 6·77-s + 15·79-s + 12·83-s − 120·85-s + 4·89-s + ⋯ |
| L(s) = 1 | − 2.23·5-s − 1.13·7-s + 0.603·11-s − 0.832·13-s + 5.82·17-s + 1.37·19-s + 17/5·25-s + 0.185·29-s − 0.538·31-s + 2.53·35-s − 0.986·37-s − 3.43·41-s − 0.457·43-s + 1.31·47-s + 3/7·49-s + 4.94·53-s − 1.34·55-s + 1.17·59-s + 0.768·61-s + 1.86·65-s + 2.13·71-s + 0.702·73-s − 0.683·77-s + 1.68·79-s + 1.31·83-s − 13.0·85-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.856147745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.856147745\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + T + T^{2} )^{3} \) |
| good | 5 | \( 1 + p T + 8 T^{2} + 7 T^{3} + 9 T^{4} - 62 T^{5} - 299 T^{6} - 62 p T^{7} + 9 p^{2} T^{8} + 7 p^{3} T^{9} + 8 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 T - 10 T^{2} - 34 T^{3} + 48 T^{4} + 416 T^{5} + 31 T^{6} + 416 p T^{7} + 48 p^{2} T^{8} - 34 p^{3} T^{9} - 10 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \) |
| 17 | \( ( 1 - 12 T + 90 T^{2} - 435 T^{3} + 90 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 36 T^{2} - 18 T^{3} + 468 T^{4} + 324 T^{5} - 5393 T^{6} + 324 p T^{7} + 468 p^{2} T^{8} - 18 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 758 p T^{7} + 4425 p^{2} T^{8} + 31 p^{3} T^{9} - 82 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 3 T - 60 T^{2} - 219 T^{3} + 1983 T^{4} + 4746 T^{5} - 51289 T^{6} + 4746 p T^{7} + 1983 p^{2} T^{8} - 219 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 22 T + 206 T^{2} + 1802 T^{3} + 18432 T^{4} + 135116 T^{5} + 808243 T^{6} + 135116 p T^{7} + 18432 p^{2} T^{8} + 1802 p^{3} T^{9} + 206 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 3 T - 54 T^{2} - 569 T^{3} + 123 T^{4} + 13170 T^{5} + 115347 T^{6} + 13170 p T^{7} + 123 p^{2} T^{8} - 569 p^{3} T^{9} - 54 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 3438 p T^{7} - 2433 p^{2} T^{8} + 531 p^{3} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 18 T + 234 T^{2} - 1917 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 9 T - 90 T^{2} + 459 T^{3} + 10161 T^{4} - 20556 T^{5} - 598421 T^{6} - 20556 p T^{7} + 10161 p^{2} T^{8} + 459 p^{3} T^{9} - 90 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 6 T - 126 T^{2} + 358 T^{3} + 12372 T^{4} - 11472 T^{5} - 838653 T^{6} - 11472 p T^{7} + 12372 p^{2} T^{8} + 358 p^{3} T^{9} - 126 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 6 T^{2} - 1366 T^{3} + 438 T^{4} - 4098 T^{5} + 1065603 T^{6} - 4098 p T^{7} + 438 p^{2} T^{8} - 1366 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 3 T + 51 T^{2} - 681 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 - 15 T + 36 T^{2} + 367 T^{3} - 3225 T^{4} + 51726 T^{5} - 676905 T^{6} + 51726 p T^{7} - 3225 p^{2} T^{8} + 367 p^{3} T^{9} + 36 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 12 T - 144 T^{2} + 582 T^{3} + 34812 T^{4} - 90444 T^{5} - 2656433 T^{6} - 90444 p T^{7} + 34812 p^{2} T^{8} + 582 p^{3} T^{9} - 144 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 2 T + 116 T^{2} - 735 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 78504 p T^{7} + 14223 p^{2} T^{8} + 573 p^{3} T^{9} - 168 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.29319534976527947596865612974, −4.15292042216568529658043095214, −4.04094944146420526592716527578, −3.92776839640958451897583665166, −3.92682316270510290491657835480, −3.81882291575814922938370235794, −3.63565590071105167766920919343, −3.53395986844692356720238544089, −3.27646062555408079185391564558, −3.24040134576854126308159665591, −3.14870129668995182039097300607, −2.90770674311356464086694725163, −2.78031442007244614784697978302, −2.66740827258942787969007912191, −2.57448810091211612169585302298, −2.10818628782220362473010350539, −1.93147905814714645674785952660, −1.81554668926605472739836447786, −1.50737773355295422680364263169, −1.25889491061855332483886152606, −0.962313580383277017699458466818, −0.923179596162115159141153386716, −0.849027982994484041888054815674, −0.50596554448087163993437101572, −0.29483179020966090786849656840,
0.29483179020966090786849656840, 0.50596554448087163993437101572, 0.849027982994484041888054815674, 0.923179596162115159141153386716, 0.962313580383277017699458466818, 1.25889491061855332483886152606, 1.50737773355295422680364263169, 1.81554668926605472739836447786, 1.93147905814714645674785952660, 2.10818628782220362473010350539, 2.57448810091211612169585302298, 2.66740827258942787969007912191, 2.78031442007244614784697978302, 2.90770674311356464086694725163, 3.14870129668995182039097300607, 3.24040134576854126308159665591, 3.27646062555408079185391564558, 3.53395986844692356720238544089, 3.63565590071105167766920919343, 3.81882291575814922938370235794, 3.92682316270510290491657835480, 3.92776839640958451897583665166, 4.04094944146420526592716527578, 4.15292042216568529658043095214, 4.29319534976527947596865612974
Plot not available for L-functions of degree greater than 10.
